Search Results for Series

Biographies

1. Zygmund biography
   
   • Aleksander Rajchman was interested in the theory of trigonometric series.
     
   • This gave Zygmund a life-long interest in the trigonometric series.
     
   • from the University of Warsaw in 1923 for a dissertation on the Riemannian theory of trigonometric series written under Aleksander Rajchman's supervision.
     
   • Among other topics, he worked on summability of numerical series, summability of general orthogonal series, trigonometric integrals, sets of uniqueness, summability of Fourier series, differentiability of functions, smooth functions, approximation theory, absolutely convergent Fourier series, multipliers and translation invariant operators, conjugate series and Taylor series, lacunary trigonometric series, series of independent random variables, random trigonometric series, the Littlewood-Paley, Luzin and Marcinkiewicz functions, boundary values of analytic and harmonic functions, singular integrals, partial differential equations and interpolation operators.
     
   • He studied topics such as Riemann summability, differentiability properties of trigonometric series and sets of uniqueness.
     
   • For example in 1926 he published six papers in Mathematische Zeitschrift in French: Contribution a l'unicite du developpement trigonometrique; Sur la theorie riemannienne des series trigonometriques; Sur la possibilite d'appliquer la methode de Riemann aux series trigonometriques sommables par le procede de Poisson; Sur les series trigonometriques sommables par le procede de Poisson; Sur un theoreme de la theorie de la sommabilite and Une remarque sur un theoreme de M Kaczmarz.
     
   • The joint Zygmund-Paley work played an important role in Zygmund's book Trigonometric Series (1935).
     
   • Each volume of the series [Monografie Mat.] published so far represents an important event in the development of mathematical research, and the present volume in this respect is second to none of its predecessors.
     
   • If one looks through the long list of books on Fourier series one can not help feeling that even the bulkiest of them are far from giving an adequate picture of the present status of the field.
     
   • The non-existence of a monograph giving such a picture was very badly felt not only by beginners but also by specialists, and the failure of so many attempts to write a real book on Fourier series created an impression that the task was almost hopeless.
     
   • The author of the present monograph completely succeeded in dispelling this "inferiority complex" and produced a book which not only introduces the reader into the immense field the theory of Fourier series but at the same time almost imperceptibly brings him to the latest achievements, many of them being due author himself.
     
   • Surely, Antoni Zygmund's "Trigonometric series" has been, and continues to be, one of the most influential books in the history of mathematical analysis.
     
   • Generations of mathematicians from Hardy and Littlewood to recent classes of graduate students specializing in analysis have viewed "Trigonometric series" with enormous admiration and have profited greatly from reading it.
     
   • In light of the importance of Antoni Zygmund as a mathematician and of the impact of "Trigonometric series", it is only fitting that a brief discussion of his life and mathematics accompany the present volume, and this is what I have attempted to give here.
     
   • The general part of the book is followed by three chapters, one on entire and meromorphic functions, one on elliptic functions, and one on G(s), Z(s) and Dirichlet series.
     
   • The purpose of these notes is to present those aspects of trigonometric interpolation which resemble the theory of Fourier series.
     

2. Stirling biography
   
   • The terms of the Snell Exhibitions is described in [James Stirling: this about series and such things (Edinburgh, 1988).',3)">3]:- .
     
   • Tweddle [James Stirling: this about series and such things (Edinburgh, 1988).',3)">3] notes that a student with the name 'James Stirling' matriculated at the University of Edinburgh on 24 March 1710, did not graduate, and has a signature which is similar to that of the mathematician.
     
   • This book is a treatise on infinite series, summation, interpolation and quadrature.
     
   • One of the main aims of the book was to consider methods of speeding up the convergence of series.
     
   • As an example of the problem he is trying to solve Stirling gives the example of the series ∑ 1/[2n(2n-1)] which had been studied by Brouncker in his work on the area under a hyperbola.
     
   • .if anyone would find an accurate value of this series to nine places ..
     
   • they would require one thousand million of terms; and this series converges much swifter than many others..
     
   • For example he defined the series Tn+1 = nTn with T1 = 1.
     
   • However, before doing so we will look at a correspondence that Stirling had with Euler since this relates to the work we have just discussed on series.
     
   • 10 (1957), 117-158.',7)">7] or [James Stirling: this about series and such things (Edinburgh, 1988).',3)">3]):- .
     
   • the more I have learned from your excellent articles, which I have seen here and there in your Transactions, concerning the nature of series, a study in which I have indeed expended much effort, the more I have wished to become acquainted with you in order that I could receive more from you yourself and also submit my own deliberations to your judgement.
     
   • Now that I have read through it diligently, I am truly astonished at the great abundance of excellent methods contained in such a small volume, by means of which you show how to sum slowly converging series with ease and how to interpolate progressions which are very difficult to deal with.
     
   • But especially pleasing to me was proposition XIV of part 1 in which you give a method by which series, whose law of progression is not even established, may be summed with great ease using only the relation of the last terms, certainly this method extends very widely and is of the greatest use.
     
   • 10 (1957), 117-158.',7)">7] or [James Stirling: this about series and such things (Edinburgh, 1988).',3)">3]):- .
     
   • He certainly did not give up mathematics when he took up the post in the mining company, and in [James Stirling: this about series and such things (Edinburgh, 1988).',3)">3] there is a discussion of unpublished mathematical work in notebooks of Stirling that were probably written between 1730 and 1745.
     
   • he surveyed the Clyde with a view to rendering it navigable by a series of locks, thus taking the first step towards making Glasgow the commercial capital of Scotland.
     
   • As Stirling's unpublished manuscripts show [James Stirling: this about series and such things (Edinburgh, 1988).',3)">3], he did go much further than the 1735 paper but probably the pressure of work at the mining company gave him too little time to polish the work.
     

3. Madhava biography
   
   • Madhava discovered the series equivalent to the Maclaurin expansions of sin x, cos x, and arctan x around 1400, which is over two hundred years before they were rediscovered in Europe.
     
   • Jyesthadeva describes Madhava's series as follows:- .
     
   • This is a remarkable passage describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion.
     
   • Perhaps we should write down in modern symbols exactly what the series is that Madhava has found.
     
   • Thus the series is .
     
   • which is equivalent to Gregory's series .
     
   • Now Madhava put q = π/4 into his series to obtain .
     
   • and he also put θ = π/6 into his series to obtain .
     
   • which can be obtained from the last of Madhava's series above by taking 21 terms.
     
   • Perhaps even more impressive is the fact that Madhava gave a remainder term for his series which improved the approximation.
     
   • He improved the approximation of the series for π/4 by adding a correction term Rn to obtain .
     
   • It is thought that the way that he found these highly accurate tables was to use the equivalent of the series expansions .
     
   • Jyesthadeva in Yukti-Bhasa gave an explanation of how Madhava found his series expansions around 1400 which are equivalent to these modern versions rediscovered by Newton around 1676.
     

4. Menshov biography
   
   • Menshov's first degree was awarded in 1916 for the thesis which he wrote on The Riemann theory of trigonometric series which was examined by Egorov and Luzin.
     
   • However, only three weeks after he graduated, Menshov discovered one of his most fundamental results on the uniqueness problem for trigonometric series.
     
   • Consider the trigonometric series .
     
   • Cantor had proved that if this series converges to 0 for all x in [0, 2π] - E, for a countable set E, then an = bn = 0 for all n.
     
   • Vallee Poussin had proved that if the above series converged to a finite Lebesgue integrable function f (x) then the given series is the Fourier series of f (x).
     
   • The remarkable, and unexpected, result that Menshov discovered in 1916 was that this was not so, for he constructed a trigonometric series which converges to 0 for all x in [0, 2π] - E, for a set E of measure zero, yet not all the coefficients of the trigonometric series are zero.
     
   • His scientific interests relate principally to the theory of trigonometric series, the theory of orthogonal series and the problem of monogenity of functions of a complex variable.
     
   • For his work on the representation of functions by trigonometric series, Menshov was awarded a State Prize in 1951.
     
   • In 1958 Menshov attended the International Congress of Mathematicians in Edinburgh and he was invited to address the Congress with his paper On the convergence of trigonometric series.
     

5. Ramanujan biography
   
   • He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.
     
   • In 1900 he began to work on his own on mathematics summing geometric and arithmetic series.
     
   • He investigated the series ∑(1/n) and calculated Euler's constant to 15 decimal places.
     
   • He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series.
     
   • Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908.
     
   • Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan's results on divergent series.
     
   • The recommendation to Ramanujan that he read Bromwich's Theory of infinite series did not please Ramanujan much.
     
   • I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as 'startling'.
     
   • Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function.
     
   • Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series.
     
   • Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic.
     

6. Euler biography
   
   • This was to find a closed form for the sum of the infinite series ζ(2) = ∑ (1/n2), a problem which had defeated many of the top mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli.
     
   • In 1737 he proved the connection of the zeta function with the series of prime numbers giving the famous relation .
     
   • Other work done by Euler on infinite series included the introduction of his famous Euler's constant γ, in 1735, which he showed to be the limit of .
     
   • Euler also studied Fourier series and in 1744 he was the first to express an algebraic function by such a series when he gave the result .
     
   • Euler wrote to James Stirling on 8 June 1736 telling him about his results on summing reciprocals of powers, the harmonic series and Euler's constant and other results on series.
     
   • Concerning the summation of very slowly converging series, in the past year I have lectured to our Academy on a special method of which I have given the sums of very many series sufficiently accurately and with very little effort.
     
   • he has two theorems for summing series by means of derivatives of the terms, one of which is the self-same result that you sent me.
     
   • I have very little desire for anything to be detracted from the fame of the celebrated Mr Maclaurin since he probably came upon the same theorem for summing series before me, and consequently deserves to be named as its first discoverer.
     
   • He considered linear equations with constant coefficients, second order differential equations with variable coefficients, power series solutions of differential equations, a method of variation of constants, integrating factors, a method of approximating solutions, and many others.
     

7. Mengoli biography
   
   • Mengoli used infinite series to good effect in Novae quadraturae arithmeticae, seu de additione fractionum published in Bologna in 1650, developing ideas which had first been investigated by Cataldi.
     
   • He begins with the summation of geometric series, then shows that the harmonic series does not converge.
     
   • In doing so he became the first person to prove that it was possible for a series whose terms tend to zero to be made larger than any given number.
     
   • He also investigated the harmonic series with alternating signs which he proved converges to log(2).
     
   • This series was also investigated by Nicolaus Mercator.
     
   • Other interesting results about series in Novae quadraturae arithmeticae include a study of the sum of reciprocals of the triangular numbers n(n+1)/2.
     
   • He then argued that the difference 1 - n/n+2 could be made smaller than any given positive number by taking n large enough so the sum of the series was 1.
     
   • , 10 Mengoli showed that the series whose nth term is 1/n(n+r) converges having sum S where .
     
   • He also showed that the series whose nth term was 1/n(n+1)(n+2) converges with sum 1/4.
     
   • However, perhaps not surprisingly, he failed to be able to sum the series whose nth term is 1/n2.
     
   • He defines limits of positive variable quantities using ideas that he had used in looking at limits of series.
     

8. Lax Anneli biography
   
   • The NML was intended as a series of monographs on various mathematical topics.
     
   • The monographs were written by individual mathematicians most of whom had not written for the high school level prior to their work in the series.
     
   • Instead, books are still being published in the NML series, though at a slower pace than during its height in the 1960s.
     
   • Lax was at the centre of the MAA's publication program for thirty-three years, overseeing the NML series.
     
   • The NML series was planned by Lax and the editorial board to [',5)">5]:- .
     
   • Lax was a skilled editor and strove to bring out the best work of the mathematicians who wrote for the series.
     
   • Around 1960 Anneli approached me about contributing a volume for an upcoming series called The New Mathematical Library that she was editing and that was designed to overcome this reluctance to write mathematical texts for students.
     
   • After much persuasion, I agreed and wrote 'The Lore of Large Numbers', Number 6 in the series currently still in print, but horribly out of date! .
     
   • There are now also more than 35 books in the NML series.
     
   • Many of Lax's admirers thought the NML should be re-titled 'ANML,' Anneli's New Mathematical Library, because of her care in developing and sustaining the series.
     
   • No other person in the history of the Association's book publishing effort has played a larger role in developing and nurturing a book series.
     

9. Lebesgue biography
   
   • Lebesgue wrote two monographs Lecons sur l'integration et la recherche des fonctions primitives (1904) and Lecons sur les series trigonometriques (1906) which arose from these two lecture courses and served to make his important ideas more widely known.
     
   • Fourier had assumed that for bounded functions term by term integration of an infinite series representing the function was possible.
     
   • From this he was able to prove that if a function was representable by a trigonometric series then this series is necessarily its Fourier series.
     
   • There is a problem here, namely that a function which is not Riemann integrable may be represented as a uniformly bounded series of Riemann integrable functions.
     
   • In 1905 Lebesgue gave a deep discussion of the various conditions Lipschitz and Jordan had used in order to ensure that a function f (x) is the sum of its Fourier series.
     
   • What Lebesgue was able to show was that term by term integration of a uniformly bounded series of Lebesgue integrable functions was always valid.
     
   • This now meant that Fourier's proof that if a function was representable by a trigonometric series then this series is necessarily its Fourier series became valid, since it could now be founded on a correct result regarding term by term integration of series.
     

10. Taylor biography
    
    • In particular they discussed infinite series and probability.
      
    • His life, however, suffered a series of personal tragedies beginning around 1721.
      
    • Taylor added to mathematics a new branch now called the "calculus of finite differences", invented integration by parts, and discovered the celebrated series known as Taylor's expansion.
      
    • It was, wrote Taylor, due to a comment that Machin made in Child's Coffeehouse when he had commented on using "Sir Isaac Newton's series" to solve Kepler's problem, and also using "Dr Halley's method of extracting roots" of polynomial equations.
      
    • Taylor initially derived the version which occurs as Proposition 11 as a generalisation of Halley's method of approximating roots of the Kepler equation, but soon discovered that it was a consequence of the Bernoulli series.
      
    • The second version occurs as Corollary 2 to Proposition 7 and was thought of as a method of expanding solutions of fluxional equations in infinite series.
      
    • The differences in Newton's ideas of Taylor series and those of Gregory are discussed in [Istor.-Mat.
      
    • The term "Taylor's series" seems to have used for the first time by Lhuilier in 1786.
      
    • Taylor series .
      
    • series for cosine .
      
    • series for sine .
      

11. Salem biography
    
    • Whenever he had free time in the evenings he worked on Fourier series, a topic which interested him throughout his life.
      
    • He became attracted to Fourier series, and the interest in the subject remained undiminished throughout his life.
      
    • Although he read some of the current literature on Fourier series, he apparently worked all alone ..
      
    • He did have some connections with Paris mathematicians, however, particularly with Denjoy who may have influenced him towards Fourier series.
      
    • He was in the right place to carry on with his interest in Fourier series, and he collaborated on this topic with Norbert Wiener and Zygmund (with whom he wrote joint papers).
      
    • Zygmund, reviewing [Oeuvres mathematique de Raphael Salem (Paris, 1967).',2)">2], puts Salem's contributions to Fourier series into perspective:- .
      
    • For the last few decades two problems were central in the field: convergence almost everywhere of Fourier series and the nature of the sets of uniqueness for trigonometric series.
      
    • Another direction in which [Salem] did a lot was applications of the calculus of probability to Fourier series and, curiously enough, this has connection with problems of uniqueness.
      
    • Moreover, it seem that, far from being incidental, as it might have appeared some 30 or so years ago, the calculus of probability is becoming a standard method of attacking problems of trigonometric series.
      
    • After Salem died his wife established an international prize for outstanding contributions to Fourier series [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
      

12. Riesz Marcel biography
    
    • He studied at Budapest University and, influenced by Fejer, undertook research on problems from the theory of series.
      
    • In it he gave the correct generalisation of Cantor's uniqueness theorem for convergent trigonometric series to trigonometric series summable by the Cesaro method.
      
    • The first period of his work, from the beginning of his doctoral research up to around the beginning of World War I, concentrated on the theory of series, in particular the summability theory of power series, trigonometric series and Dirichlet series.
      
    • Another highlight from this period is his beautiful proof of Fatou's theorem which give conditions under which the power series of an analytic function converges to a point on its circle of convergence.
      
    • In a joint work with Hardy The general theory of Dirichlet's series, published by Cambridge University Press in 1915, he introduced Riesz means.
      
    • He gave an important series of lectures Clifford numbers and spinors at the University of Maryland between October 1957 and January 1958.
      

13. Mandelbrojt biography
    
    • He remained in the United States as a lecturer at the Rice Institute in Houston during 1926-27 and published Modern researches on the singularities of functions defined by Taylor's series in a Rice Institute Pamphlet (1927).
      
    • Mandelbrojt published an important book Series de Fourier et classes quasi-analytiques de fonctions in 1935.
      
    • The present lectures give an excellent account of the modern theory of classes of infinitely differentiable functions of a real variable and may be regarded as a second edition of the author's book "Series de Fourier et classes quasi-analytiques de fonctions" to include the work of the author and of Henri Cartan since that date.
      
    • Continuing to work at the Rice Institute during World War II, Mandelbrojt continued to publish important work In 1944 he published a series of lectures he had given at the Institute under the title Dirichlet series:- .
      
    • This monograph consists of a series of lectures delivered by the author and is not a complete treatment of general Dirichlet series.
      
    • He continued as association with the Rice Institute, howver, and continued to publish work in their Rice Institute Pamphlet series.
      
    • He published in that series General theorems of closure in 1951 which presented results concerning closure of translations and closure of linear combinations of derivatives of a function.
      
    • In the following year he published Series adherentes, regularisation des suites, applications in Paris, then in 1958 Composition theorems in the Rice Institute Series.
      
    • various density functions for sets of positive integers, adherent series, the Fourier-Carleman transform and the spectrum of a function defined by means of this transform.
      
    • In 1969 Mandelbrojt published Series de Dirichlet.
      

14. Nilakantha biography
    
    • The series π/4 = 1 - 1/3 + 1/5 - 1/7 + ..
      
    • is a special case of the series representation for arctan, namely .
      
    • It is well known that one simply puts x = 1 to obtain the series for π/4.
      
    • 63 (5) (1990), 291-306.',4)">4] reports on the appearance of these series in the work of Leibniz and James Gregory from the 1670s.
      
    • The contributions of the two European mathematicians to this series are well known but in [Math.
      
    • 63 (5) (1990), 291-306.',4)">4] the results on this series in the work of Madhava nearly three hundred years earlier as presented by Nilakantha in the Tantrasamgraha is also discussed.
      
    • Nilakantha derived the series expansion .
      
    • An interesting feature of his work was his introduction of several additional series for π/4 that converged more rapidly than .
      
    • 63 (5) (1990), 291-306.',4)">4] provides a reconstruction of how he may have arrived at these results based on the assumption that he possessed a certain continued fraction representation for the tail series .
      

15. Steinhaus biography
    
    • Steinhaus studied Lebesgue's two major books Lecons sur l'integration et la recherche des fonctions primitives (1904) and Lecons sur les series trigonmetriques (1906) around 1912 after completing his doctorate.
      
    • by Steinhaus and Banach, concentrated mainly on functional analysis and its diverse applications, the general theory of orthogonal series, and probability theory.
      
    • Another important publishing venture in which Steinhaus was involved, begun in 1931, was a new series of Mathematical Monographs.
      
    • The series was set up under the editorship of Steinhaus and Banach from Lvov and Knaster, Kuratowski, Mazurkiewicz, and Sierpinski from Warsaw.
      
    • An important contribution to the series was a volume written by Steinhaus jointly with Kaczmarz in 1937, The theory of orthogonal series.
      
    • Some of Steinhaus's early work was on trigonometric series.
      
    • He gave an example of a trigonometric series which diverged at every point, yet its coefficients tended to zero.
      
    • He also gave an example of a trigonometric series which converged in one interval but diverged in a second interval.
      
    • As we have noted above, other contributions by Steinhaus were on orthogonal series, probability theory, real functions and their applications.
      

16. Machin biography
    
    • This series (among others for the same purpose, and drawn from the same principle) I received from the excellent analyst, and my much esteemed friend Mr John Machin; and by means thereof, van Ceulen's number, or that in Art.
      
    • No indication is given in Jones's work, however, as to how Machin discovered his series expansion for π so when de Moivre wrote to Johann Bernoulli on 8 July 1706 telling him about Machin's series for π he suggested that Johann Bernoulli might tell Jakob Hermann about Machin's unproved result.
      
    • He did so and Hermann quickly discovered a proof that Machin's series converges to π.
      
    • He produced techniques that show other similar series also converge rapidly to π and he wrote on 21 August 1706 to Leibniz giving details.
      
    • Two years later, on 6 July 1708, de Moivre wrote again to Johann Bernoulli about Machin's series, on this occasion giving two different proofs that it converged to π.
      
    • Machin had explained to Taylor in Child's Coffeehouse how to use Newton's series to solve Kepler's problem and also how Halley's method finds roots of polynomial equations.
      
    • Mr William Jones's Synopsis palmariorum matheseo which was published in the year 1706 is the only book in which as I believe the series of Mr Machin had ever made its appearance before the publication of my Dissertation on the Use of the Negative Sign in Algebra in the year 1758.
      
    • Machin's work on the series for has proved of lasting importance, but most of his other contributions are not of the same high standard.
      

17. Rey Pastor biography
    
    • Between 1911 and 1916, La Junta Para Ampliacion de Estudios funded Rey Pastor to carry out a series of visits to Germany.
      
    • However he was not one to remain fixed in one place for a long time and went to Barcelona in 1915 to give a series of lectures at the Institut d'Estudia.
      
    • It dealt with the study of the method of summation of series.
      
    • This article by Rey Pastor is framed by a long series of works, begun at the beginning of the twentieth century, on problems of summing series, convergence algorithms, singular integrals and comparative studies of series and integrals.
      
    • He first presented his work in this area in 1926 in his course on "Series and Integrals" which he gave at Buenos Aires University.
      
    • He continued working on problems related to the theory of summation of divergent series throughout the 1930's and published much of his work in international journals.
      

18. Carleson biography
    
    • In 1966 Carleson solved one of the outstanding problems of mathematics in his paper On convergence and growth of partial sums of Fourier series.
      
    • Fourier, in 1807, had claimed that every function equals the sum of its Fourier series.
      
    • A major research area throughout the 19th century concerned the convergence of Fourier series, and continuous functions whose Fourier series diverges everywhere were constructed by du Bois-Reymond.
      
    • In 1913 Luzin conjectured that if a function f is square integrable then the Fourier series of f converges pointwise to f Lebesgue almost everywhere.
      
    • Carleson lectured on his spectacular result at the International Congress of Mathematicians at Moscow in 1966 when he gave the address Convergence and summability of Fourier series.
      
    • The citation emphasizes not only Carleson's fundamental scientific contributions, the best known of which perhaps are the proof of Luzin's conjecture on the convergence of Fourier series, the solutions of the corona problem and the interpolation problem for bounded analytic functions, the solution of the extension problem for quasiconformal mappings in higher dimensions, and the proof of the existence of 'strange attractors' in the Henon family of planar maps, but also his outstanding role as scientific leader and advisor.
      
    • His research in analysis is a series of towering and fundamental discoveries.
      

19. Fine Nathan biography
    
    • who will present a series of at most three lectures accessible to a large fraction of those who teach college mathematics.
      
    • He is perhaps best known for his book Basic hypergeometric series and applications published in the Mathematical Surveys and Monographs Series of the American Mathematical Society.
      
    • Fine was at that time engaged in his own special development of q-hypergeometric series, and as the years passed he kept adding to his results and polishing his presentation.
      
    • We became somewhat diverted while looking at Fine's text Basic hypergeometric series and applications when we began to look at Andrews' Introduction.
      
    • For far too long, there has been a dearth of good references on basic hypergeometric series.
      
    • The present book and Basic hypergeometric series by G Gasper and M Rahman have appeared in the past two years to greatly rectify this situation.
      
    • This is a very personal book, a distillation of those results in basic hypergeometric series which hold the most appeal to its author.
      

20. Dirichlet biography
    
    • These papers introduce Dirichlet series and determine, among other things, the formula for the class number for quadratic forms.
      
    • He turned to Laplace's problem of proving the stability of the solar system and produced an analysis which avoided the problem of using series expansion with quadratic and higher terms disregarded.
      
    • Dirichlet is also well known for his papers on conditions for the convergence of trigonometric series and the use of the series to represent arbitrary functions.
      
    • These series had been used previously by Fourier in solving differential equations.
      
    • Earlier work by Poisson on the convergence of Fourier series was shown to be non-rigorous by Cauchy.
      
    • Because of this work Dirichlet is considered the founder of the theory of Fourier series.
      
    • Riemann, who was a student of Dirichlet, wrote in the introduction to his habilitation thesis on Fourier series that it was Dirichlet [Mathematics in Berlin (Berlin, 1998), 33-40.',11)">11]:- .
      

21. Dahlquist biography
    
    • Three methods, old but not so well known, transform an infinite series into a complex integral over an infinite interval.
      
    • Applications are made to slowly convergent alternating and positive series, to Fourier series, to the numerical analytic continuation of power series outside the circle of convergence, and to ill-conditioned power series.
      
    • [The second part] is mainly concerned with the derivation, analysis and applications of a summation formula, due to Lindelof, for alternating series and complex power series, including ill-conditioned power series.
      

22. Bosanquet biography
    
    • Bosanquet's early 1930s papers were on series and integration, a 1930 paper being on fractional integration, a topic he would return to many times.
      
    • For example he was a visiting professor at the University of Utah during 1964-65 where he gave a major lecture series on The history and development of the theory of divergent series and integrals.
      
    • During 1969-70 he visited the University of Western Ontario and gave another major lecture series, this time on Matrix transformations and sequence spaces with applications to summability.
      
    • Bosanquet wrote many papers on the convergence and summability of Fourier series.
      
    • He also wrote on the convergence and summability of Dirichlet series and studied specific kinds of summability such as summability factors for Cesaro means.
      
    • He saw Hardy's great book 'Divergent Series' through the press during Hardy's last illness and he later edited the volume on 'Series' in Hardy's Collected Works; he was chief editor for the last two of the seven volumes.
      

23. Li Shanlan biography
    
    • Xu may have been a governor but he was also an excellent mathematician with an interest in infinite series.
      
    • He produced his own versions of logarithms, infinite series, and combinatorics but he did not follow the style of western mathematics but made his research naturally develop out of the foundations of Chinese mathematics.
      
    • Li wrote Duoji bilei (Summing finite series) (published in 1867 as part of his collected works) where, in Chapter 4, he gave fascinating formulae relating binomial coefficients, Stirling numbers, Eulerian numbers and many others.
      
    • The summation of series constitutes a branch of Chinese mathematics called Short Width [Chapter 4 of the Nine Chapters on the Mathematical Art.
      
    • The works of the great astronomer Guo Shoujing concerning the inequalities of the solar and lunar motion, Wang Lai's iterated sums, Dong Fangli's cyclotomical computations, and lastly the summation of series which appear in the algebra and the differential calculus of the Westerners constitute the major part of this chapter.
      
    • Zhu Shijie from the Yuan dynasty is the only one who has made use of the prescriptions relating to summation of series.
      
    • But his intention was only to expound the algebra and for that reason he presents the summation of series neither precisely nor methodically.
      

24. Wolf biography
    
    • Wolf wrote on prime number theory and geometry, then later on probability and statistics - a series of papers discussed Buffon's needle experiment in which he estimated π by Monte Carlo methods.
      
    • I decided to carry out corresponding series of tests, hoping to obtain, not , but at least new proofs about the rules governing a finite number of trials.
      
    • On a plate of about one square foot I drew a series of parallels at a distance of 45 mm, and from a knitting needle I broke a piece of 36 mm length - thus getting as close as 1/100 to the ideal ratio according to the instruction above.
      
    • Thus I obtained in the first series of trials for 100 tosses a mean of 21.76 ± 0.64 tosses in which the needle intersected the parallels.
      
    • In the second series of trials I obtained 71.34 ± 1.25.
      
    • In the third series of trials I obtained 50.64 ± 0.83.
      
    • a number that lies within the error limits of the mean from the third series of trials.
      

25. Goldbach biography
    
    • When Bernoulli started to discuss infinite series with Goldbach as they talked in Oxford, Goldbach confessed that he knew nothing about the topic.
      
    • Bernoulli gave him a loan of a book on the topic by his uncle Jacob Bernoulli but Goldbach found infinite series too difficult at that time, and gave up his attempts to understand Jacob Bernoulli's text.
      
    • We mentioned that Goldbach gave up his attempts to understand infinite series in 1712.
      
    • However in 1717 he read an article by Leibniz on computing the area of a circle and this led him to look again at the theory of infinite series.
      
    • We should, however, mention his another two of his papers on infinite series De transformatione serierum (1729) and De terminis generalibus serierum (1732).
      
    • The first of these introduced a method of transforming one series into another while the sum of the series remains fixed.
      

26. Bernoulli Jacob biography
    
    • Jacob Bernoulli returned to Switzerland and taught mechanics at the University in Basel from 1683, giving a series of important lectures on the mechanics of solids and liquids.
      
    • By 1689 he had published important work on infinite series and published his law of large numbers in probability theory.
      
    • Jacob Bernoulli published five treatises on infinite series between 1682 and 1704.
      
    • Euler was the first to find the sum of this series in 1737.
      
    • Bernoulli also studied the exponential series which came out of examining compound interest.
      
    • The Bernoulli numbers appear in the book in a discussion of the exponential series.
      
    • Bernoulli greatly advanced algebra, the infinitesimal calculus, the calculus of variations, mechanics, the theory of series, and the theory of probability.
      

27. Lexis biography
    
    • From 1876-79 Lexis studied data presented as a series over time.
      
    • He initiated the study of time series with his 1879 article On the theory of the stability of statistical series.
      
    • Lexis required a binomial dispersion for his series to be stable and this ruled out most interesting series.
      
    • After what is not much more than a three year period working on statistics in Freiburg, he began to produce a series of papers on economics.
      
    • In addition to studying economics, the theory of population, and statistics, he also worked on the production of an economic encyclopaedia, edited a series of works on education in general and university education in particular, and was the director of the first institute of actuarial science in Germany.
      

28. Du Bois-Reymond biography
    
    • The standard technique to solve partial differential equations used Fourier series but Cauchy, Abel and Dirichlet had all pointed out problems associated with the convergence of the Fourier series of an arbitrary function.
      
    • In 1873 du Bois-Reymond was the first person to give an example of a continuous function whose Fourier series diverges at a point.
      
    • Perhaps what was even more surprising, the Fourier series of du Bois-Reymond function diverged at a dense set of points.
      
    • The important work Eine neue Theorie der Convergenz und Divergenz von Reihen mit positiven Gliedern ("A new theory of convergence and divergence of series with positive terms") led to an increasing understanding of the whole concept of a function.
      
    • The conception of space as static and unchanging can never generate the notion of a sharply defined, uniform line from a series of points however dense, for, after all, points are devoid of size, and hence no matter how dense a series of points may be, it can never become an interval, which must always be regarded as the sum of intervals between points.
      

29. Cantor biography
    
    • This was due to Heine, one of his senior colleagues at Halle, who challenged Cantor to prove the open problem on the uniqueness of representation of a function as a trigonometric series.
      
    • He published further papers between 1870 and 1872 dealing with trigonometric series and these all show the influence of Weierstrass's teaching.
      
    • Cantor published a paper on trigonometric series in 1872 in which he defined irrational numbers in terms of convergent sequences of rational numbers.
      
    • Between 1879 and 1884 Cantor published a series of six papers in Mathematische Annalen designed to provide a basic introduction to set theory.
      
    • Soon Cantor was publishing in Mittag-Leffler's journal Acta Mathematica but his important series of six papers in Mathematische Annalen also continued to appear.
      
    • The fifth paper in this series Grundlagen einer allgemeinen Mannigfaltigkeitslehre was also published as a separate monograph and was especially important for a number of reasons.
      

30. Durell biography
    
    • He was promoted to senior mathematics master at Winchester College in 1910 and began publishing a series of articles in the Mathematical Gazette.
      
    • After the end of the war he returned to Winchester College and began publishing a series of articles in the Mathematical Gazette and a remarkable series of textbooks which would make him the best known writer of English school mathematics texts.
      
    • All chapters conclude with a series of exercises, with solutions at the end of the book.
      
    • Contents include the properties of the triangle and the quadrilateral; equations, sub-multiple angles, and inverse functions; hyperbolic, logarithmic, and exponential functions; and expansions in power-series.
      
    • Further topics encompass the special hyperbolic functions; projection and finite series; complex numbers; de Moivre's theorem and its applications; one- and many-valued functions of a complex variable; and roots of equations.
      

31. Maclaurin biography
    
    • It is in the Treatise of fluxions that Maclaurin uses the special case of Taylor's series now named after him and for which he is undoubtedly best remembered today.
      
    • The Maclaurin series was not an idea discovered independently of the more general result of Taylor for Maclaurin acknowledges Taylor's contribution.
      
    • Another important result given by Maclaurin, which has not been named after him or any other mathematician, is the important integral test for the convergence of an infinite series.
      
    • Maclaurin series .
      
    • Maclaurin series for cosine .
      
    • Maclaurin series for sine .
      

32. Bronowski biography
    
    • He continued his research in geometry publishing a series of papers On triple planes and a paper The figure of six points in space of four dimensions.
      
    • [I am a] mathematician trained in physics, who was taken into the life sciences in middle age by a series of lucky chances.
      
    • He had presented a series for BBC television in the early 1960s called Insight in which he had looked at mathematical ideas such as probability, scientific ideas such as entropy and also the extent of human intelligence.
      
    • His last major project was to write and narrate the BBC television series The Ascent of Man which was filmed between July 1971 and December 1972.
      
    • The thirteen part series was broadcast in 1973 and also published in book form in that year.
      
    • I [EFR] remember watching this remarkable television series which, in a style much copied since, described in humanist terms the work of Pythagoras, Newton, Einstein, Galen, Versalius, Darwin, Mendel, Szilard, John von Neumann and many others showing their contributions as significant highlights in human development.
      

33. Jordan biography
    
    • Indeed Jordan introduced the concept of a composition series (a series of subgroups each normal in the preceding with the property that no further terms could be added to the series so that it retains that property).
      
    • The composition factors of a group G are the groups obtained by computing the factor groups of adjacent groups in the composition series.
      
    • Jordan proved the Jordan-Holder theorem, namely that although groups can have different composition series, the set of composition factors is an invariant of the group.
      
    • Among Jordan's many contributions to analysis we should also mention his generalisation of the criteria for the convergence of a Fourier series.
      

34. Kaluznin biography
    
    • He worked for the CNRS, published a series of papers on the structure of Sylow p-subgroups of symmetric groups, and in 1948 defended his doctoral thesis on the same topic.
      
    • His activities and achivements during the decade 1960-1970 includes: conducting research, teaching at Kiev State University and the Kiev Pedagogical Institute, consulting for the Department of Mathematical Linguistics, serving as a senior researcher at the Institute of Cybernetics of the Ukrainian Academy of Sciences, organising series of public lectures on mathematics, and serving as a member on editorial boards of several scientific journals.
      
    • Using his techniques, he was able to describe the characteristic subgroups of the Sylow p-subgroups, their derived series, their upper and lower central series, and more.
      
    • A particularly important result is the well-known theorem of Krasner and Kaluznin concerning the embeddings of a group with a subnormal series into the wreath product of the factors of the series.
      

35. Edmonds biography
    
    • Two papers appeared in 1942: On the multiplication of series which are infinite in both directions was published in the Journal of the London Mathematical Society while On the Parseval formulae for Fourier transforms appeared in the Proceedings of the Cambridge Philosophical Society.
      
    • In the first of these papers Edmonds looks at the doubly infinite series ∑ an where the sum is over both positive and negative integers.
      
    • The series is said to converge if the two series, one defined over the positive integers, the other defined over the negative integers, both converge.
      
    • In this case the original series is said to have sum A = A' + A'' where A' and A'' are the sums over the positive and negative integers respectively.
      
    • In a series of papers published over the following years Edmonds examined a whole variety of different conditions on the functions f and g which give the required equalities.
      

36. Weierstrass biography
    
    • The transformation of his conception of an analytic function from a differentiable function to a function expansible into a convergent power series was made during this early period of Weierstrass's mathematical activity.
      
    • This paper did not give the full theory of inversion of hyperelliptic integrals that Weierstrass had developed but rather gave a preliminary description of his methods involving representing abelian functions as constantly converging power series.
      
    • The topics of his lectures included:- the application of Fourier series and integrals to mathematical physics (1856/57), an introduction to the theory of analytic functions (where he set out results he had obtained in 1841 but never published), the theory of elliptic functions (his main research topic), and applications to problems in geometry and mechanics.
      
    • Its contents were: numbers, the function concept with Weierstrass's power series approach, continuity and differentiability, analytic continuation, points of singularity, analytic functions of several variables, in particular Weierstrass's "preparation theorem", and contour integrals.
      
    • The standards of rigour that Weierstrass set, defining, for example, irrational numbers as limits of convergent series, strongly affected the future of mathematics.
      
    • Known as the father of modern analysis, Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations.
      

37. James Ralph biography
    
    • In certain problems (for example, in the theory of uniqueness of trigonometric series) it is desirable to define a second integral of a given function without defining the first integral.
      
    • The main result of the present paper is that the sum of an everywhere convergent trigonometric series is integrable in the proposed sense, and the series itself is the Fourier series of the sum.
      
    • Some later papers continuing this theme were Generalized nth primitives (1954), Integrals and summable trigonometric series (1955), and Summable trigonometric series (1956).
      

38. James Ralph biography
    
    • In certain problems (for example, in the theory of uniqueness of trigonometric series) it is desirable to define a second integral of a given function without defining the first integral.
      
    • The main result of the present paper is that the sum of an everywhere convergent trigonometric series is integrable in the proposed sense, and the series itself is the Fourier series of the sum.
      
    • Some later papers continuing this theme were Generalized nth primitives (1954), Integrals and summable trigonometric series (1955), and Summable trigonometric series (1956).
      

39. Murnaghan biography
    
    • He also published The orthogonal and symplectic groups in 1958 which arose from a series of twenty lectures he gave in Dublin in 1957.
      
    • It covers topics such as: vectors and matrices; Fourier series; boundary value problems; Legendre and Bessel functions; integral equations; the calculus of variations and dynamics; and the operational calculus.
      
    • The first of these is a short book of less that 100 pages written for engineers and scientists, while the second consists of 19 lectures on such topics as: the Fourier integral; the Laplace integral transformation; the differential equations of Laguerre and Bessel; properties of special functions; asymptotic series for an error function, and for certain Bessel functions.
      
    • The 'converging factor' for an asymptotic series representing a function f(x) is that number by which the (n+1)st term of the series must be multiplied so that the result of adding this product to the sum of the first n terms will be f(x).
      
    • This report describes the determination to high precision of this factor for the asymptotic series representing the probability integral.
      

40. Zaanen biography
    
    • As a student, he came into contact with the ideas of modern analysis via Zygmund's book on trigonometric series and Banach's book on linear transformations.
      
    • During the war years he published papers such as On some orthogonal systems of functions (1939), A theorem on a certain orthogonal series and its conjugate series (1940), On some orthogonal systems of functions (1940), Uber die Existenz der Eigenfunktionen eines symmetrisierbaren Kernes (1942), Uber vollstetige symmetrische und symmetrisierbare Operatoren (1943), Transformations in Hilbert space which depend upon one parameter (1944), and On the absolute convergence of Fourier series (1945).
      
    • The results appeared in 1946 and 1947 in a series of seven papers in the Proceedings of the Royal Academy of Sciences of The Netherlands.
      
    • Their series of sixteen notes (all published in the Nederl.
      

41. James Ralph biography
    

42. Van Vleck biography
    
    • For example he published On the determination of a series of Sturm's functions by the calculation of a single determinant (1899), On linear criteria for the determination of the radius of convergence of a power series (1900), On the convergence of continued fractions with complex elements (1901), A determination of the number of real and imaginary roots of the hypergeometric series (1902), On an extension of the 1894 memoir of Stieltjes (1903), and On the extension of a theorem of Poincare for difference-equations (1912).
      
    • Van Vleck was American Mathematical Society Colloquium lecturer in 1903 giving six lectures on divergent series and continued fractions.
      
    • He published these lectures in the first volume of the series American Mathematical Society Colloquium Publications.
      

43. Denjoy biography
    
    • In 1934 he wrote that his greatest achievements had been the integration of derivatives, the computation of the coefficients of a converging trigonometric series, a theorem on quasi-analytic functions, and differential equations on a torus.
      
    • The second of these topics, computation of the coefficients of a converging trigonometric series, was the subject of a four volume work Lectures on the computation of coefficients in a trigonometric series which appeared between 1941 and 1949.
      
    • These four volumes were an expanded version of work which had appeared in a series of papers by Denjoy beginning in 1920.
      
    • However, Choquet describes the four volume work Lectures on the computation of coefficients in a trigonometric series which contains the famous Denjoy integral, as [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
      

44. Julia biography
    
    • Volume 2, in three parts, consists of articles on (i) J points of functions of one variable, (ii) J points of functions of several variables, and (iii) Series of iterates.
      
    • Volume 3 contains four parts: (i) Functional equations and conformal mapping; (ii) Conformal mapping; (iii) General lectures; and (iv) Isolated works in analysis on Implicit function defined by the vanishing of an active function, and on certain series.
      
    • An admirable account of the theory of Fourier series (pp.
      
    • This book is the sixteenth of the well known series, 'Cahiers Scientifiques,' and is the first of a series which proposes to give the mathematical foundation of quantum mechanics.
      

45. Pade biography
    
    • In his thesis Pade made the first systematic study of what we call today Pade approximants, which are rational approximations to functions given by their power series.
      
    • At around the same time Euler used Pade-type approximation to find the sum of a series.
      
    • The method continued to be used from time to time by various mathematicians, for example Kummer in 1837 used Pade approximants to sum series which only converged very slowly.
      
    • Pade established various properties of this table in his thesis and developed the ideas further in later papers, particularly in 1899 when he studies the exponential series and in 1901 when he considered (1+x)m, for m not an integer.
      
    • Although the theory of Pade approximants which he had developed in his thesis, and in many later papers, was not quick to be taken up by many other mathematicians, it did become well known after Borel presented Pade approximants in his 1901 book on divergent series.
      

46. Borel biography
    
    • the theory of measure, Borel's theory of divergent series, his theory of non-analytic continuation and the theory of quasi-analytic functions all derive from ideas which make their first appearance in this paper.
      
    • In [Enseignement mathematique 11 (1965), 1-95.',8)">8] Borel's mathematical work is divided into the following topics: Arithmetic; Numerical series; Set theory; Measure of sets; Rarefaction of a set of measure zero; Real functions of real variables; Complex functions of complex variables; Differential equations; Geometry; Calculus of probabilities; and Mathematical physics.
      
    • Borel, although not the first to define the sum of a divergent series, was the first to develop a systematic theory for a divergent series which he did in 1899.
      
    • In addition, between 1921 and 1927, Borel published a series of papers on game theory and became the first to define games of strategy.
      
    • He published many outstanding works over the years including Lecons sur la theorie des fonctions (1898), Lecons sur les series entieres (1900), Lecons sur les fonctions divergentes (1901), Lecons sur les fonctions de variables reelles et les developpements en series de polynomes (1905), Le Hasard (1913), Lecons sur les fonctions monogenes uniformes d'une variable complexe (1917), L'Espace et le temps (1921), Methodes et problemes de la theorie des fonctions (1922), L'espace et le temps (1922), Traite du calcul des probabilites et ses applications (1924-1934), and Principes et formules classiques du calcul des probabilites (1925).
      

47. Kuttner biography
    
    • Fourier series, strong summability, Riesz means, Norland methods, and Tauberian theory.
      
    • Most of Kuttner's early work is on Fourier series and summability.
      
    • Hardy quotes some of these early results of Kuttner's in his treatise Divergent series (1949).
      
    • at the age of only 26, Kuttner proved a basic theorem in the general theory of trigonometric series, a result delightful for both the deceptive simplicity of its statement and the elegance of its proof.
      
    • Zygmund greatly admired this theorem of Kuttner, which now occupies an honoured place in Zygmund's monumental work on trigonometric series.
      

48. Gregory biography
    
    • He visited Flanders, Rome and Paris on his journey but spent most time at the University of Padua where he worked on using infinite convergent series to find the areas of the circle and hyperbola.
      
    • We can now be certain that during the summer of 1668 Gregory was completely familiar with the series expansions of sin, cos and tan.
      
    • Although he did not disclose his methods in the small treatise he discussed topics including various series expansions, the integral of the logarithmic function, and other related ideas.
      
    • For his reluctance to publish his "several universal methods in geometry and analysis" when he heard through Collins of Newton's own advances in calculus and infinite series, he postumously paid a heavy price ..
      
    • However, we now summarise these and other contributions in the hope that, despite his reluctance to publish his methods, his remarkable contributions might indeed be more widely understood: Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as 1670; he discovered Taylor expansions more than 40 years before Taylor; he solved Kepler's famous problem of how to divide a semicircle by a straight line through a given point of the diameter in a given ratio (his method was to apply Taylor series to the general cycloid); he gives one of the earliest examples of a comparison test for convergence, essentially giving Cauchy's ratio test, together with an understanding of the remainder; he gave a definition of the integral which is essentially as general as that given by Riemann; his understanding of all solutions to a differential equation, including singular solutions, is impressive; he appears to be the first to attempt to prove that π and e are not the solution of algebraic equations; he knew how to express the sum of the nth powers of the roots of an algebraic equation in terms of the coefficients; and a remark in his last letter to Collins suggests that he had begun to realise that algebraic equations of degree greater than four could not be solved by radicals.
      

49. Escher biography
    
    • He later tried working with the concept of similarities, using identical motifs of diminishing size, arranged in a series of concentric circles, but as with much of his work, he was unhappy about the final quality.
      
    • Escher used pictures to tell a story in his Metamorphosis series of designs.
      
    • These designs brought together many of Escher's skills and show the transformation from one distinct object to another, by means of a series of slight changes to a regular pattern in the plane.
      
    • An Italian coastline is transformed through a series of convex polygons into a regular pattern in the plane until finally a distinct, coloured, human motif emerges, signifying his change of perspective from landscape work to regular division of the plane.
      
    • Escher fell ill initially in 1964 whilst delivering a series of lectures in North America.
      

50. Seidenberg biography
    
    • He held a Visiting Professorship at the University of Milan and he gave several series of lectures there.
      
    • In fact he was in Milan in the middle of giving a lecture series at the time of his death.
      
    • His career included a Guggenheim Fellowship [awarded 1953], visiting Professorships at Harvard and at the University of Milan, and numerous invited addresses, including several series of lectures at the University of Milan, the National University of Mexico, and at the Accademia dei Lincei in Rome.
      
    • At the time of his death, he was in the midst of another series of lectures at the University of Milan.
      
    • Concepts such as plane curve, intersection multiplicity, branch, genus, and linear series are introduced in a concrete, computational way; the necessary abstract algebra is kept in a secondary position whenever possible.
      

51. Levinson biography
    
    • The turning point in Levinson's studies had come when he signed up for Wiener's graduate course on Fourier series and integrals in 1933-34.
      
    • In 1940 Levinson published Gap and density theorems in the American Mathematical Society Colloquium Publication Series.
      
    • It was a great tribute to the young mathematician that he had been invited to write a book in a series which was reserved for distinguished senior mathematicians.
      
    • Levinson wrote only two papers on time series, but these had a large impact.
      
    • Shortly before his death he wrote a series of important papers on the Riemann hypothesis arising from this fundamental number theory paper.
      

52. Kronecker biography
    
    • students to hear that Kronecker was questioned at his oral on a wide range of topics including the theory of probability as applied to astronomical observations, the theory of definite integrals, series and differential equations, as well as on Greek, and the history of philosophy.
      
    • In 1870 Heine published a paper On trigonometric series in Crelle's Journal, but Kronecker had tried to persuade Heine to withdraw the paper.
      
    • Even the concept of an infinite series, for example one which increases according to definite powers of variables, is in my opinion only permissible with the reservation that in every special case, on the basis of the arithmetic laws of constructing terms (or coefficients), ..
      
    • certain assumptions must be shown to hold which are applicable to the series like finite expressions, and which thus make the extension beyond the concept of a finite series really unnecessary.
      

53. Carlson biography
    
    • thesis On a class of Taylor series on 26 May 1914 (undertaken with the supervision of Anders Wiman), and became Doctor of Philosophy on 30 May 1914.
      
    • Some of his most well-known contributions are a theorem connected to the Phragmen-Lindelof principle, a theorem about the zeros of the V-function and several theorems about power series with integer coefficients.
      
    • Such names as Carlson inequality, Carlson - Levin constants, Carlson theorem in complex analysis, Polya - Carlson theorem on rational functions and Carlson theorem on Dirichlet series are well-known in mathematics (see [Inequalities (Cambridge, 1934).
      
    • Carlson, in a series of papers, investigated Dirichlet series and proved in 1922 that if f (z) = sigman1inf ann-z is convergent in Re z ≥ 0 and bounded in every Re z > δ > 0, then, for each > 0, .
      

54. Wronski biography
    
    • He criticised Lagrange's use of infinite series and introduced his own ideas for series expansions of a function.
      
    • Out of this came his "universal Hoene-Wronski series" or "la serie universelle de Wronski".
      
    • This consisted of the development of a function as a series in terms of arbitrary functions.
      
    • The coefficients in this series are determinants now known as Wronskians (so named by Muir in 1882).
      

55. Lindelof biography
    
    • Then he worked on analytic functions, applying results of Mittag-Leffler in a study of the asymptotic investigation of Taylor series.
      
    • He considered analogues of Fourier series and applied them to gamma functions.
      
    • Moreover he considers series analogous to Fourier summation formulas and applications to the gamma function and the Riemann function.
      
    • In addition, new results concerning the Stirling series and analytic continuation are presented.
      
    • The book concludes with an asymptotic investigation of series defined by Taylor's formula.
      

56. Jyesthadeva biography
    
    • Jyesthadeva describes Madhava's series as follows:- .
      
    • This is a remarkable passage describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion.
      
    • To see how this description of the series fits with Gregory's series for arctan(x) see the biography of Madhava.
      

57. Morin biography
    
    • These memoirs presented the results of a series of carefully executed experiments on friction which he began planning in 1829.
      
    • During 1853-56 Morin undertook a series of experiments on the resistance of building materials which he published in a series of papers.
      
    • a totalizing anemometer, one of a series which was to be set into chimneys in the new buildings of the Palais de Justice in order to monitor ventilation.
      
    • The revolutions were counted through a series of gear wheels, so as to indicate the volume of air passing through the shaft in a period of time.
      

58. Rajagopal biography
    
    • Rajagopal studied sequences, series, summability.
      
    • He also studied functions of a complex variable giving an analogue of a theorem of Edmund Landau on partial sums of Fourier series.
      
    • In several papers he studied the relation between the growth of the mean values of an entire function and that of its Dirichlet series.
      
    • He showed that the series for tan-1x discovered by Gregory and those for sin x and cos x discovered by Newton were known to the Hindus 150 years earlier.
      
    • He identified the Hindu mathematician Madhava as the first discoverer of these series.
      

59. Tauber biography
    
    • He lectured in Vienna on the theory of series, trigonometric series, and potential theory.
      
    • Tauber's lack of success in being given a professorial position was certainly not due to any lack of mathematical ability, for he continued to publish a series of high quality papers.
      
    • He obtained important results on divergent series and the name 'Tauberian Theorems' was coined by Hardy and Littlewood.
      
    • The conditions which Tauber gave to allow him to prove the converse of Abel's limit theorem on power series are now known as 'Tauberian conditions' and appeared in Ein Satz aus der Theorie der unendlichen Reihen (1897).
      

60. Szasz biography
    
    • Szasz's main work was in real analysis, particularly Fourier series.
      
    • His most important contributions are probably between 1915 and 1930 when he made a series of remarkable contributions to a number of different areas.
      
    • Other work by Szasz made major contributions to questions posed by Landau on the maximum modulus of the partial sums of a power series.
      
    • He also studied problems on power series related to work of Frigyes Riesz.
      
    • Some of Szasz's contributions to Fourier series related to results proved by Bernstein, Hardy, Littlewood and Fejer.
      

61. Fejer biography
    
    • In 1900 Fejer published a fundamental summation theorem for Fourier series.
      
    • However there were problems regarding his appointment to the chair as is recounted in [Functions, series, operators, Colloq.
      
    • He worked on power series and on potential theory.
      
    • Much of his work is on Fourier series and their singularities but he also contributed to approximation theory.
      
    • One of Fejer's students described his lecturing style in the following way (see [Functions, series, operators, Colloq.
      

62. Holder biography
    
    • Otto Holder worked on the convergence of Fourier series and in 1884 he discovered the inequality now named after him.
      
    • He became interested in group theory through Kronecker and Klein and proved the uniqueness of the factor groups in a composition series.
      
    • Holder became a lecturer at Gottingen in 1884 and at first he worked on the convergence of Fourier series.
      
    • He began to study the Galois theory of equations and from there he was led to study compostion series of groups.
      
    • Holder proved the uniqueness of the factor groups in a composition series, the theorem now called the Jordan-Holder theorem.
      

63. Gudermann biography
    
    • His own contributions tended to be a whole series of special cases (although this could not have been obvious at the time) which were forgotten later when the general results which included them were found.
      
    • Gudermann, at this time, was particularly interested in the theory of elliptic functions and in the expansion of functions by power series.
      
    • In particular his use of power series in the study of the hyperbolic functions is of importance.
      
    • The transformation of his conception of an analytic function from a differentiable function to a function expandable into a convergent power series was made during this early period of Weierstrass's mathematical activity.
      

64. Bonnet biography
    
    • One year before this, in 1843, Bonnet had written a paper on the convergence of series with positive terms.
      
    • Another paper on series in 1849 was to earn him an award from the Brussels Academy.
      
    • However between these two papers on series, Bonnet had begun his work on differential geometry in 1844.
      
    • Between 1844 and 1867 he published a series of papers on the differential geometry of surfaces.
      

65. Lorenz Edward biography
    
    • The generalized vorticity equation is satisfied by formal infinite series representing the density and wind fields.
      
    • The first few terms of a particular series solution are obtained explicitly.
      
    • The series appear to converge near the north pole, and determine a model of a polar air mass.
      
    • He used this series of three lectures as a basis for his famous text The essence of chaos (1993).
      

66. Cunha biography
    
    • De Pombal put through a series of major reforms, and around 1758-59 he moved against the Jesuits and the Society of Jesus.
      
    • Da Cunha develops a criterion for the convergence of a series which he uses to define the exponential function in a rather modern way, and from these develops the binomial series.
      
    • In Principios Matematicos da Cunha also gave a definition of the convergence of a series which is equivalent to Cauchy's convergence criterion.
      

67. Ramanathan biography
    
    • After a series of papers published in North American journals such as Identities and congruences of Ramanujan type (1950), The Theory of Units of Quadratic and Hermitian Forms (1951), Abelian quadratic forms (1952), and Units of quadratic forms (1952), he returned to publishing his research in Indian journals.
      
    • For several years, Professor Ramanathan had been actively interested in the study of published and unpublished work of Srinivasa Ramanujan, expounding, elucidating and extending Ramanujan's beautiful work on singular values of certain modular functions, Rogers-Ramanujan continued fractions and hypergeometric series.
      
    • An example of a paper motivated by his study of Ramanujan's work is Hypergeometric series and continued fractions (1987).
      
    • The author extends much of the work mentioned above to basic hypergeometric series.
      

68. Laurent Hermann biography
    
    • However he was already working on writing mathematics texts, his first being Traite des series in 1862.
      
    • He wrote 30 books and a fair number of papers on infinite series, equations, differential equations and geometry.
      
    • N Ya Sanin and A V Letnikov published a series of papers in Matem.
      
    • Particularly given the topics on which Laurent worked, it is easy to assume that Laurent series must be named after him.
      
    • However this is not the case and Laurent series are in fact named after Pierre Laurent.
      

69. Luke biography
    
    • Not only did he use rational approximation, but Luke also developed series expansions as an approximation method.
      
    • For example he expanded hypergeometric functions in series of Laguerre and Hermite polynomials.
      
    • He gave a wonderful series of lectures on special functions, asymptotic analysis, and approximation theory.
      
    • While at MRI he gave an extensive series of lectures on the history of philosophy, focusing especially on Spinoza, whose work he believed, contains the most meaningful elements of those ethical and intellectual ideals which alone can provide a personal bedrock in an uncertain, frenetically changing world.
      

70. Haar biography
    
    • Haar asked a series of fundamental questions about systems of orthonormal functions on the interval [0, 1].
      
    • Haar wrote: one wants to be able to determine sufficient conditions that a series of such functions is convergent; one wants examples of relatively sensible functions which do not converge in the pointwise or uniform sense; one wants to understand how summation methods may be used to overcome the problems of divergence; and one wants to know exactly when, if the series of partial sums of an orthogonal expansion of a function converges, its limit equals the original function.
      
    • He constructed what is now known as Haar's orthonormal basis to answer the question of divergence of continuous functions expanded as series of orthonormal systems of functions.
      

71. Brink biography
    
    • He then entered Harvard University where he was awarded his doctorate in 1916 for his thesis Some Integral Tests for the Convergence and Divergence of Infinite Series.
      
    • Finally we give some examples of Brink's papers: A new integral test for the convergence and divergence of infinite series (1918); A new sequence of integral tests for the convergence and divergence of infinite series (1919); The May Meeting of the Minnesota Section (1927); Recent Publications: Reviews: Studies in the History of Statistical Method - With Special Reference to Certain Education Problems (1929); The May Meeting of the Minnesota Section (1930); A Simplified Integral Test for the Convergence of Infinite Series (1931); Recent Publications: Reviews: Differential Equations (1932); The Annual Meeting of the Minnesota Section (1937); and College Mathematics During Reconstruction (1944).
      

72. Polya biography
    
    • While the book was being worked on, Polya continued a remarkable series of publications, with a total of 31 papers appearing during the three years 1926-28.
      
    • For example, in 1918 he published papers on series, number theory, combinatorics and voting systems.
      
    • Polya's interest in complex analysis led him to investigate singularities of power series, gap theorems, power series with integral coefficients and those taking integral values at the positive integers, the Polya representation for entire functions of exponential type, and the location of zeros.
      

73. Jarnik biography
    
    • He studied the problem for the particular case of the ellipsoid in a series of papers.
      
    • During the decade 1939-49 he wrote a series of papers dealing with the geometry of numbers, in particular dealing with Minkowski's inequality for convex bodies.
      
    • He also wrote on rearrangement of infinite series, trigonometric series and other areas of analysis.
      

74. Gronwall biography
    
    • Gronwall's work contains classical analysis (Fourier series, Gibbs phenomenon, summability theory, Laplace and Legendre series), differential and integral equations, analytic number theory (transcendental numbers, divisor function, L-function of Dirichlet), complex function theory (Dirichlet L-series, conformal mappings, univalent functions), differential geometry, mathematical physics (problems of elasticity, ballistics, induction, potential theory, kinetic theory of gases, optics), nomography, atomic physics (wave mechanics of hydrogen and helium atom, lattice theory of crystals) and physical chemistry where he is especially known as a very important contributor.
      
    • The Gronwall summation method of series appeared in 1932 as a generalization of de la Vallee Poussin and Cesaro summation methods (cf.
      

75. Mackey biography
    
    • He then produced a series of important papers on group representations including On induced representations of groups (1951), Induced representations of locally compact groups (1952), and Symmetric and anti symmetric Kronecker squares and intertwining numbers of induced representations of finite groups (1953).
      
    • During the academic year 1966-67, Mackey delivered a series of lectures on group representations and their applications at Oxford University in England where he was George Eastman visiting professor.
      
    • Mackey has written many beautiful survey articles and in 1992 the American Mathematical Society and the London Mathematical Society in their wonderful series 'History of Mathematics' published The scope and history of commutative and noncommutative harmonic analysis by Mackey.
      
    • 1968), and Caroline Series (Ph.D.
      

76. Kendall Maurice biography
    
    • Kendall continued a remarkable stream of research papers on topics such as the theory of k-statistics, time series, and rank correlation methods and a monograph Rank Correlation in 1948.
      
    • In 1963 he published (jointly with P A P Moran) Geometrical probability followed by Time series (1973) in which Kendall states his objectives to bridge the gap between "sophisticated theory and practical applications" in the field of time series and to "treat the subject in its entirety for the benefit of the practising statistician".
      
    • He also published A course in multivariate analysis and Cluster analysis as well a whole series of articles Studies in the history of probability and statistics.
      

77. Keen biography
    
    • In the 1980s she collaborated with Caroline Series on work on Riemann surfaces which returned to ideas that she had studied for her doctoral thesis:- .
      
    • Using powerful techniques developed by Thurston that involve hyperbolic three-manifolds, Keen and Series gave a geometric interpretation to Maskit's parameters.
      
    • Much of Keen's work has been done in collaborations with other mathematicians and after her work with Caroline Series she teamed up with Paul Blanchard, Robert Devaney, and Lisa Goldberg to produce important results on dynamical systems.
      
    • She teamed up again with Caroline Series for their joint article Pleating invariants for punctured torus groups (2004).
      
    • Keen, working with Nikola Lakic, wrote the book Hyperbolic geometry from a local viewpoint which was published in 2007 by Cambridge University Press in the London Mathematical Society Student Texts series.
      
    • The Emmy Noether Lectures are a prestigious series of lectures organised by the Association for Women in Mathematics.
      

78. Newcomb biography
    
    • He wrote a paper showing how the coordinates of a planet might be represented by trigonometric series.
      
    • Laplace had devised a method involving cosine series for computing the perturbing force on a planet caused by other planets.
      
    • The coefficients in the series were known as 'Laplace coefficients' but the drawback of the method was that it only worked for circular orbits.
      
    • Newcomb showed how to extend Laplace's series to give a perturbing function in the case of elliptical orbits by introducing differential operators which act on the Laplace coefficients.
      

79. Renyi biography
    
    • started regularly to attend a lecture series I held to June of 1942.
      
    • from the University of Szeged, with Frigyes Riesz as his thesis advisor, for a thesis on Cauchy-Fourier series.
      
    • Results from his doctoral thesis appeared in the paper On the summability of Cauchy-Fourier series (1950).
      
    • The first result, regarding the representation of an even number, is an approximation to the unproved Goldbach conjecture and supersedes an earlier proof of the same proposition by Estermann (1932) which made use of an unproved generalized Riemann hypothesis for all Dirichlet L-series.
      

80. Northcott biography
    
    • He then went on to study for Part III, taking courses on Fourier series and divergent series from G H Hardy which strongly influenced him.
      
    • He wrote up the results he had obtained while in hospital in India as the paper Abstract Tauberian theorems with applications to power series and Hilbert series which was published in 1947.
      

81. Bocher biography
    
    • He was awarded a doctorate in 1891 for his dissertation Uber die Reihenentwicklungen der Potentialtheorie (Development of the Potential Function into Series) having been encouraged to study this topic by Klein who acted as supervisor.
      
    • Bocher published around 100 papers on differential equations, series, and algebra.
      
    • In a seventy page article in 1906, Introduction to the theory of Fourier's series published in the Annals of Mathematics, he gave the first satisfactory treatment of the Gibbs phenomenon (he wrote another paper On Gibbs' phenomenon in 1914).
      
    • Bocher was honoured by the American Mathematical Society when he was chosen to give the first series of Colloquium lectures in 1896.
      

82. Orlicz biography
    
    • In 1928 he wrote his doctoral thesis Some problems in the theory of orthogonal series under the supervision of Eustachy Zylinski.
      
    • His book Linear Functional Analysis, (Peking 1963, 138 pp - in Chinese), based on a series of lectures delivered in German on selected topics of functional analysis at the Institute of Mathematics of Academia Sinica in Beijing in 1958, was translated into English and published in 1992 by World Scientific, Singapore.
      
    • Orlicz's contribution is important in the following areas in mathematics: function spaces (mainly Orlicz spaces), orthogonal series, unconditional convergence in Banach spaces, summability, vector-valued functions, metric locally convex spaces, Saks spaces, real functions, measure theory and integration, polynomial operators and modular spaces.
      
    • For example, the Orlicz-Pettis theorem says that in Banach spaces the classes of weakly subseries convergent and norm unconditionally convergent series coincide.
      

83. Herstein biography
    
    • Noncommutative rings appeared in The Carus Mathematical Monographs series published by The Mathematical Association of America.
      
    • This colourful and informative book on noncommutative ring theory is based on a series of expository lectures given by the author in the summer of 1965 at Bowdoin College before an audience of teachers from colleges and small universities.
      
    • The spirit of the Carus Monograph series is clearly embodied in this moving and excellently written account of important aspects of classical and modern ring theory.
      
    • Topics in Ring Theory was based on lectures Herstein gave at the University of Chicago and first published in the University of Chicago Mathematics Lecture Notes series.
      

84. Green Sandy biography
    
    • By that time M H A Newman's plan to use specially designed electronic computers to assist in the decipherment of the "Fish" series of coded messages was well advanced.
      
    • Finally let us mention Sandy's little book Sequences and series (1958) whose aim is stated in the Preface:- .
      
    • We also mention the lectures given by Sandy at Groups St Andrews 1989 when he was a main speaker giving a series of lectures on Schur algebras and general linear groups.
      
    • Green: "Sequences and Series" .
      

85. Wright biography
    
    • The resulting financial disaster caused Kate and Maitland to separate and Kate Wright, being an excellent musician and trained music teacher, took up a series of positions at boarding schools in the south of England.
      
    • Also among his early work was a series of three papers titled Asymptotic partition formulae.
      
    • The third in the series Asymptotic partition formulae, III.
      
    • These topics are: prime numbers; congruences and the quadratic reciprocity law; continued fractions; irrational, algebraic and transcendental numbers; quadratic fields; arithmetical functions, their order of magnitude and the Dirichlet or power series which generate them; partitions and representations of numbers as sums of squares, cubes and higher powers; Diophantine approximation; and the geometry of numbers.
      

86. De Bruijn biography
    
    • He continued to hold this position until June 1944 [Applied Logic Series 28 (Kluwer Academic Publishers, Dordrecht, 2003).',1)">1]:- .
      
    • Also in 1943, in addition to his doctoral thesis, he published On the absolute convergence of Dirichlet series, On the number of solutions of the system ..
      
    • He began publishing papers on combinatorics relevant to his work during this period such as The problem of optimum antenna current distribution (1946), A combinatorial problem (1946), On the zeros of a polynomial and of its derivative (1946), and A note on van der Pol's equation (1946) [Applied Logic Series 28 (Kluwer Academic Publishers, Dordrecht, 2003).',1)">1]:- .
      
    • The Preface [Applied Logic Series 28 (Kluwer Academic Publishers, Dordrecht, 2003).',1)">1] records:- .
      

87. Lidstone biography
    
    • He read the paper Note on the Summation of a Trigonometrical Series to the Society at its meeting on Friday 10 March 1922.
      
    • This paper is composed of a series of commentaries, illustrated frequently with numerical examples, on the subject of interpolation.
      
    • that the corresponding Charlier type B series ..
      
    • The author shows by numerical examples that in many cases this series gives a better fit than Pearson's type III curves.
      

88. Kummer biography
    
    • He published a paper on hypergeometric series in Crelle's Journal in 1836 and he sent a copy of the paper to Jacobi.
      
    • This resulted in a series of sympathetic revolutions against the governments of the German Confederation.
      
    • He extended Gauss's work on hypergeometric series, giving developments that are useful in the theory of differential equations.
      
    • He was the first to compute the monodromy groups of these series.
      

89. Ritt biography
    
    • One year before the publication of this work, Ritt had published Theory of Functions which provides an introduction to the theory of functions in a series of short and to the point lecture notes giving the student an account of the fundamental definitions and theorems of the subject.
      
    • He produced two books on the subject which contained the results from a long series of papers which he produced on the topic.
      
    • In a remarkable series of papers which appeared in the Annals of Mathematics he investigated a differential group of order n was he defined as a power series, with certain properties, in two sets of n indeterminates and their derivatives of various orders.
      

90. Hindenburg biography
    
    • His first papers on mathematics were published in 1776 when he studied series.
      
    • Hindenburg published a series of works on combinatorial mathematics, in particular probability, series and formulae for higher differentials.
      
    • Gudermann, best known as the teacher of Weierstrass, worked on the expansion of functions into power series and, as shown by Manning in [Arch.
      

91. Riemann biography
    
    • He spent thirty months working on his Habilitation dissertation which was on the representability of functions by trigonometric series.
      
    • While preceding papers have shown that if a function possesses such and such a property, then it can be represented by a Fourier series, we pose the reverse question: if a function can be represented by a trigonometric series, what can one say about its behaviour.
      
    • Riemann studied the convergence of the series representation of the zeta function and found a functional equation for the zeta function.
      

92. Bari biography
    
    • Bari worked under Luzin for her doctorate on the theory of trigonometrical series.
      
    • In [Women of Mathematics (Westport, Conn., 1987), 6-12.',1)">1] her final publication, a research monograph on trigonometric series, is described as follows:- .
      
    • It has become a standard reference for mathematicians specializing in the theory of functions and the theory of trigonometric series.
      
    • Bari also wrote textbooks, Higher Algebra (1932) and The Theory of Series (1936).
      

93. Farey biography
    
    • The reason we have included him is that he made one mathematical observation and, from this, the Farey series of fractions has been named.
      
    • We shall discuss below Farey's contribution to mathematics and also look at others who contributed to Farey series.
      
    • In the second paragraph he defines the Farey series and states the "curious property".
      
    • The Farey series (really a sequence) is defined as follows.
      

94. Hirzebruch biography
    
    • In 1962 Hirzebruch gave a series of seminars at Brandeis and Berkeley.
      
    • In 1981 a series of five lectures by Hirzebruch and eight lectures by Gerard van der Geer were combined into a book by the two authors entitled Lectures on Hilbert modular surfaces.
      
    • The first 30 Arbeitstagungen which he organised form the "First Series" with a "Second Series" beginning in 1993.
      

95. Mittag-Leffler biography
    
    • In this paper Mittag-Leffler proposed a series of general topological notions on infinite point sets based on Cantor's new set theory [Mathematics and Mathematicians : Mathematics in Sweden before 1950 (Providence, R.I., 1998), 85-96.',8)">8]:- .
      
    • Between 1900 and 1905 Mittag-Leffler published a series of five papers which he called "Notes" on the summation of divergent series.
      
    • The aim of these notes was to construct the analytical continuation of a power series outside its circle of convergence.
      

96. Bromwich biography
    
    • He worked on infinite series, particularly during his time in Galway.
      
    • In 1908 he published his only large treatise An introduction to the theory of infinite series which was based on lectures on analysis he had given at Galway.
      
    • In a series of papers he put Heaviside's calculus on a rigorous basis treating the operators as contour integrals.
      
    • Thomas Bromwich's Infinite Series .
      

97. Stiefel biography
    
    • There followed a series of four papers by Stiefel and his two assistants Heinz Rutishauser and Ambros Speiser, Programmgesteuerte digitale Rechengerate (elektronische Rechenmaschinen) appearing in 1950 and 1951.
      
    • In this series of papers the authors discuss in very considerable detail a number of the important mathematical questions that naturally arise in the design of a digital computer.
      
    • The first of the series was in 1964 followed by meetings in 1967, 1969, 1972, 1975 and 1978 [Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics, Math.
      
    • (i) We develop the averaging method, based consistently on Lie series, and we deal in detail with the implications of this basic concept.
      

98. Tamarkin biography
    
    • It was published in English in Mathematische Zeitschrift in 1928 as Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions.
      
    • Tamarkin published a series of papers written jointly with Hille between 1932 and 1934.
      
    • For example they published: On the summability of Fourier series (two papers), On a theorem of Hahn-Steinhaus, On a theorem of Paley and Wiener, On the theory of linear integral equations.
      
    • He published a major text The Problem of Moments written jointly with J A Shohat in the American Mathematical Society Mathematical Surveys Series (1943).
      

99. Dini biography
    
    • Dini looked at infinite series and generalised results such as a theorem of Kummer and one of Riemann, the ideas for which had first emerged in work of Dirichlet.
      
    • He discovered a condition, now known as the Dini condition, ensuring the convergence of a Fourier series in terms of the convergence of a definite integral.
      
    • As well as trigonometric series, Dini studied results on potential theory.
      
    • He published Foundations of the theory of functions of a real variable in 1878; a treatise on Fourier series in 1880; and a two volume work Lessons on infinitesimal analysis with the first volume appearing in 1907 and the second in 1915.
      

100. Wolf Frantisek biography
     
     • His thesis Contribution a la theorie des series trigonometriques generalisees et des series a fonctions orthogonales was supervised by Otakar Borvka, and he was awarded the degree Rerum Naturum Doctor in 1928.
       
     • The first was An extension of the Phragmen-Lindelof theorem while the second was On summable trigonometrical series: an extension of uniqueness theorems.
       
     • The first gives, under certain rather complex conditions, inversion formulae for trigonometric integrals and the second asserts that under the same conditions the difference between the given trigonometric integral and the trigonometric series of a certain function will be uniformly summable to zero throughout a given interval.
       
     • In Chapter VI he gives some results on the inversion of order of integration in a trigonometric integral equivalent to the integration of trigonometric series term by term.
       

101. Beckenbach biography
     
     • He wrote a series of books and acted as editor for seveal more texts.
       
     • In 1961 he edited a second series of Modern mathematics for the engineer which was divided into three parts as the earlier volume.
       
     • Beckenbach contributed to mathematics with a series of texts for schools and colleges.
       

102. Cartwright biography
     
     • in a long series of papers she continued to explore the theory of complex (especially entire) functions; particularly their strange behaviour where they "blow up" ..
       
     • Monthly 103 (10) (1996), 833-845.',3)">3], [Obituary : Dame Mary Cartwright DBE (1900-1998) (9 April 1998, Guardian).',5)">5], and [European Mathematical Society Newsletter 30 (1999), 21-23.',6)">6] as "wry" and Caroline Series writes in [European Mathematical Society Newsletter 30 (1999), 21-23.',6)">6]:- .
       
     • I [EFR] watched the TV documentary and fully agree with Caroline Series' comment.
       

103. Laguerre biography
     
     • He found a divergent series, the first few terms of which gave a good approximation to the integral.
       
     • He went on to investigate properties of the polynomials, proving orthogonality relations and also showing that an arbitrary function could be expanded in a 'Fourier type' series in Laguerre polynomials.
       
     • That it was developed from a divergent series is especially remarkable.
       

104. Beatty biography
     
     • Another venture led by Beatty was organising a series of Mathematical Expositions to be produced by the Department.
       
     • He wrote the following as a Preface to an early volume which shows his thinking behind the series:- .
       
     • A series of books, published under the auspices of the University of Toronto and bearing the title 'Mathematical Expositions', represents an attempt to meet this need.
       

105. Bortolotti biography
     
     • Bortolotti studied topology at first but later went in the direction of analysis considering the calculus of finite differences, continued fractions, convergence of infinite algorithms, summation of series, the asymptotic behaviour of series and improper integrals.
       
     • In the 1940 paper on Babylonian mathematics, Bortolotti gives a summary of problems published by Neugebauer but argues that the fact that large series of examples for quadratic equations are made up from the same roots demonstrates that this pair of roots has an 'arcane mystic property'.
       

106. Goursat biography
     
     • He then produced an impressive series of papers which contributed to almost every area of analysis which was being studied at that time.
       
     • Goursat's papers on the theory of linear differential equations and their rational transformations, as well as his studies on hypergeometric series, Kummer's equation, and the reduction of abelian integrals form, in the words of Picard "a remarkable ensemble of works evolving naturally one from the other".
       
     • Volume 1 covers applications to geometry, expansion in series, definite integrals, and derivatives and differentials.
       
     • Despite working on the many new editions of Cours d'analyse mathematique , Goursat found time to write other texts such as Le probleme de Backlund (1925), and Lecons sur les series hypergeometriques et sur quelles fonctions qui s'y rattachent (1936).
       

107. Khinchin biography
     
     • This first paper began a series of publications by Khinchin on properties of functions which are retained after deleting a set of density zero at a given point.
       
     • It was first published in 1943 and the eight lectures it contains are: Continuum; Limits; Functions; Series; Derivative; Integral; Series expansions of functions; and Differential equations.
       

108. White biography
     
     • Klein attended the Columbian Exposition and then went to Evanston to spend two weeks there at White's invitation to give a series of lectures on contemporary mathematical research.
       
     • Now, why would it not be possible to combine with this miscellaneous program (which ought by all means to be kept up) something more akin to university models? Would not a series of three or six lectures on nearly related topics, if well chosen, prove attractive and useful to larger numbers? .
       
     • White's proposal was indeed taken up by the American Mathematical Society and the result was the American Mathematical Society Colloquium lectures which are published as the American Mathematical Society Colloquium Publication series.
       

109. Cassels biography
     
     • His mathematical publications started in about 1947 with a series of papers on the geometry of numbers, in particular papers on theorems of Khinchin and of Davenport, and on a problem of Mahler.
       
     • After further papers on Diophantine equations and Diophantine approximation he wrote a series of five papers on Some metrical theorems in Diophantine approximation.
       
     • After further papers on subgroups of infinite abelian groups and normal numbers he wrote a series of eight papers on Arithmetic on curves of genus 1.
       
     • Ian Cassels has made many distinguished contributions to the theory of numbers; possibly his most important work is on the arithmetic of elliptic curves, published in a series of papers between 1959 and 1964.
       

110. Yang Hui biography
     
     • Yang also gave formulae for the sum of certain series, for example he found the sum of the squares of the natural numbers from m2 to (m+n)2 and showed that .
       
     • 8 (1) (1981), 61-66.',14)">14] for a discussion of the geometrical ideas which lie behind Yang's approach to summing series.
       
     • The topics covered by Yang include multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures.
       

111. Douglas biography
     
     • In a series of papers from 1927 onwards Douglas worked towards the complete solution: Extremals and transversality of the general calculus of variations problem of the first order in space (1927), The general geometry of paths (1927-28), and A method of numerical solution of the problem of Plateau (1927-28).
       
     • In the 47 page text, Douglas also mentions Fourier series and transforms, Denjoy integrals and the double integrals of Riemann and of Lebesgue.
       
     • He also presented a series of papers On the basis theorem for finite abelian groups.
       

112. Stoilow biography
     
     • His work was having a major international impact and he was invited to Paris where he gave a series of lectures on his work in February 1931.
       
     • Stoilow gave a series of six lectures on Riemann surfaces at the Istituto di Alta Matematica in Rome in April, 1957.
       
     • After a fairly standard introduction to the general theory, beginning with power series, he goes on, in volume 1, to look at topics such as entire and meromorphic functions, doubly periodic functions, conformal mapping on the boundary of a Jordan region, multiple-valued functions, and applications of modular functions to the Picard circle of ideas.
       

113. Castelnuovo biography
     
     • In 1873 Alexander von Brill and Max Noether had published a joint work on properties of linear series.
       
     • In the second of these, which appeared in 1891, he gave the first systematic use of the characteristic series and of the adjoint system.
       
     • Castelnuovo produced a series of papers over a period of 20 years which, together with Enriques, finally produced a classification of algebraic surfaces.
       

114. Bohr Harald biography
     
     • Harald Bohr worked on Dirichlet series, and applied analysis to the theory of numbers.
       
     • Bohr's interest in which functions could be represented by a Dirichlet series led him to devise the theory of almost periodic functions.
       
     • The fundamental theorem for almost periodic functions is a generalisation of the Parseval identity for Fourier series.
       

115. Whittaker John biography
     
     • He was interested over many years in expanding functions in a series of polynomials and Whittaker's constant is named after him.
       
     • Interpolatory function theory (1939, reprinted 1964), Series of Polynomials (1944), and Sur les Series de Base de Polynomes Quelconques (1949).
       
     • The second book is only 43 pages long and is the result of a series of lectures given by Whittaker at the Fouad I University, Egypt in 1943.
       

116. Delsarte biography
     
     • He published a series of papers on this topic in 1934-35: Les fonctions moyenne-periodiques (1934); Application de la theorie des fonctions moyenne-periodiques a la resolution de certaines equations integrales (1934); Application de la theorie des fonctions moyenne-periodiques a la resolution des equations de Fredholm-Norlund (1935); and Les fonctions moyenne-periodiques (1935).
       
     • Delsarte worked in analysis extending work on series expansions due to Whittaker and Watson.
       
     • These works had convinced him that a good understanding of the formal properties of [series expansions of functions] was necessary to a fruitful study of their domains of definition and their mode of convergence.
       

117. Chapman biography
     
     • His first research was on summable series and he wrote two papers on this topic, one of them a joint paper with Hardy.
       
     • During the war, between 1915 and 1917, he completed a series of important papers on thermal diffusion and the fundamentals of gas dynamics.
       
     • Between 1913 and 1919 he published another important series of papers, this time on terrestrial magnetism which we comment on below.
       

118. Rademacher biography
     
     • He also wrote a number of textbooks such as Lectures on analytic number theory (1955), Lectures on elementary number theory (1964), Dedekind sums (1972), Topics in analytic number theory (1973), and Higher mathematics from an elementary point of view which was only published in 1983 but was based on a series of lectures he delivered at Stanford University in 1947.
       
     • In this remarkable series of lectures the author has taken a number of interesting mathematical threads and woven them into a colorful tapestry.
       
     • They appeared as Dedekind sums in the Carus Mathematical Monographs series in 1972.
       

119. Heine biography
     
     • Before arriving at Halle, Heine published on partial differential equations and during his first few years teaching at Halle he wrote papers on the theory of heat, summation of series, continued fractions and elliptic functions.
       
     • At Halle, Heine taught a variety of courses such as: potential theory and its applications, number theory, Fourier series, trigonometric series, mechanics, and the theory of heat.
       

120. Macdonald biography
     
     • the relations between convergent series and asymptotic expansions, the zeros and the addition theorem of the Bessel functions, various Bessel integrals, spherical harmonics and Fourier series.
       
     • Macdonald worked on electric waves and solved difficult problems regarding diffraction of these waves by summing series of Bessel functions.
       

121. Schubert biography
     
     • Schubert was editor of Sammlung Schubert, a series of textbooks.
       
     • He wrote the first in the series Arithmetik und Algebra and a later book in the series on analysis Niedere Analysis.
       

122. Galileo biography
     
     • Galileo spent three years holding this post at the university of Pisa and during this time he wrote De Motu a series of essays on the theory of motion which he never published.
       
     • From these reports, and using his own technical skills as a mathematician and as a craftsman, Galileo began to make a series of telescopes whose optical performance was much better than that of the Dutch instrument.
       
     • He made a long series of observations and was able to give accurate periods by 1612.
       

123. Meray biography
     
     • This work is a book Nouveau precis d'analyse infinitesimale which aims to present the theory of functions of a complex variable using power series.
       
     • It is another rigorous work and in fact between 1872 and 1894 Meray produced a series of papers which remove geometric considerations from analytic proofs.
       
     • Meray's work consistently follows Lagrange in basing the whole of analysis on the concept of functions written as Taylor series.
       

124. Bayes biography
     
     • Another mathematical publication on asymptotic series appeared after his death where he showed that the series for log z! given by Stirling and de Moivre, was not valid since it diverged.
       
     • This notebook contains a considerable amount of mathematical work, including discussions of probability, trigonometry, geometry, solution of equations, series, and differential calculus.
       

125. Al-Umawi biography
     
     • After describing the very briefly the basic arithmetical operations of addition and multiplication, al-Umawi moves on to discuss the summation of series.
       
     • Among the series al-Umawi considers are arithmetic and geometric series.
       

126. Faraday biography
     
     • Faraday introduced a series of six Christmas lectures for children at the Royal Institution in 1826.
       
     • He published his first paper in what was to become a series on Experimental researches on electricity in 1831.
       
     • These two final series of lectures by Faraday were published and have become classics.
       

127. Woodhouse biography
     
     • He wrote an three papers in the Philosophical Transactions of the Royal Society in 1801, 1802 and an important book Principles of Analytic Calculation in 1803 which attempted to put the calculus on a rigorous algebraic foundation using a formal series expansions method similar to that developed by Lagrange [Dictionary of National Biography (Oxford, 2004).',2)">2]:- .
       
     • In essence Woodhouse was dealing with Taylor series of a function, from which he could directly read off the first, second, third etc.
       
     • derivatives from the coefficients of the terms of the series without involving any limiting process.
       

128. Hurwitz biography
     
     • The lectures contained Weierstrass's version of the arithmetisation of analysis including his "construction" of the real numbers, the ε, δ approach to analysis and his theory of complex functions based on power series.
       
     • He also wrote several papers on Fourier series.
       
     • Hurwitz informed E Landau about Kakeya's result (corrected); Landau needed the result in a proof of a theorem on infinite power series.
       

129. Parseval biography
     
     • The first was Memoire sur la resolution des equations aux differences partielle lineaires du second ordre dated 5 May 1798, the second was Memoire sur les series et sur l'integration complete d'une equation aux differences partielle lineaires du second ordre, a coefficiens constans dated 5 April 1799, the third was Ingegration generale et complete des equations de la propogation du son, l'air etant considere avec les trois dimensions dated 5 July 1801, the fourth was Ingegration generale et complete de deux equations importantes dans la mecanique des fluides dated 16 August 1803, and finally Methode generale pour sommer, par le moyen des integrales definies, la suite donnee par le theoreme de M Lagrange, au moyen de laquelle il trouve une valeur qui satisfait a une equation algebrique ou transcendente dated 7 May 1804.
       
     • Today this theorem is seen in the context of Fourier series, and often also in more abstract settings which are quite far removed from Parseval's original ideas.
       
     • The original theorem was concerned with summing infinite series.
       
     • The improved version, as given in 1801, states that if two series .
       

130. Banach biography
     
     • Another important publishing venture, begun in 1931, was a new series of Mathematical Monographs.
       
     • The first volume in the series Theorie des Operations lineaires was written by Banach and appeared in 1932.
       
     • In addition, he contributed to measure theory, integration, the theory of sets, and orthogonal series.
       

131. Marcinkiewicz biography
     
     • Zygmund was undertaking research on trigonometric series and in 1931-32 he gave a course on this topic at Wilno for the first time.
       
     • The only visible trace of Schauder's influence is a very interesting paper of Marcinkiewicz on the multipliers of Fourier series, a paper which originated in connection with a problem proposed by Schauder ..
       
     • He suggested problems on general orthogonal systems to Marcinkiewicz and this resulted in a series of papers from him on this topic.
       

132. Bremermann biography
     
     • In fact his interest in the theory of computation went back to his days as a graduate student in Munster when he attended a series of lectures on Turing machines.
       
     • In 1978 he gave the "What Physicists Do" series of lectures at The Sonoma State University.
       
     • In this series he discussed the physical limitations to mathematical understanding of physical and biological systems.
       

133. Cesaro biography
     
     • Sur diverses questions d'arithmetique was the first of a series which Cesaro wrote on the theory of numbers.
       
     • the number of common divisors of two numerals, determination of the values of the sum totals of their squares, the probability of incommensurability of three arbitrary numbers, and so on; to these he attempted to apply obtained results in the theory of Fourier series.
       
     • He also contributed to the study of divergent series, a topic which interested him early in his career, and we should note that in his work on mathematical physics he was a staunch follower of Maxwell.
       

134. Carlyle biography
     
     • He gave a series of lectures beginning in May 1837 on the German influence on Britain, and another series in the following year on European literature.
       
     • Further lectures series were given in 1839 and 1840.
       

135. Schmetterer biography
     
     • One advantage of working in the Henschel aircraft factory was the fact that he had to use Fourier series, and now back in the Mathematical Institute he began to study these series more deeply.
       
     • In 1949 he obtained the right to teach in universities after submitting his Habilitation thesis Uber die Approximation gewisser trigonometrischer Reihen on the theory of trigonometrical series.
       

136. MacRobert biography
     
     • The E-function was a generalisation of the generalised hypergeometric functions, and from 1938 onwards MacRobert produced a whole series of works on the properties of the E-function and integrals with E-functions.
       
     • Formulae for generalized hypergeometric functions as particular cases of more general formulae (1939) showed how certain known formulae for generalized hypergeometric functions can be derived as particular cases of formulae of more general type involving multiple series; Some formulae for the E-function (1941) showed how special cases of the formulae derived lead to interesting relations between Bessel functions, Legendre functions and confluent hypergeometric functions; and Proofs of some formulae for the hypergeometric function and the E-function (1943) gave alternative proofs for some known theorems on hypergeometric functions, then gives a formula for an integral involving the product of two E-functions.
       
     • He continued to produce papers on the E-function such as On an identity involving E-functions (1948), Integral of an E-function expressed as a sum of two E-functions (1953), An integral involving an E-function and an associated Legendre functions of the first kind (1953), Integrals involving E-functions (1958), Infinite series of E-functions (1959).
       

137. Hardy biography
     
     • wrote many papers on the convergence of series and integrals and allied topics.
       
     • Hardy's interests covered many topics of pure mathematics - Diophantine analysis, summation of divergent series, Fourier series, the Riemann zeta function, and the distribution of primes.
       

138. Mercator Nicolaus biography
     
     • Mercator discovered the well known series, sometimes called Mercator's series, .
       
     • This series was also investigated by Mengoli.
       

139. Stolz biography
     
     • He later dedicated an increasing part of his research to real analysis, in particular to convergence problems in the theory of series, including double series; to the discussion of the limits of indeterminate ratios; and to integration.
       
     • He published a series of books under the title Lecons nouvelles sur l'analyse infinitesimale et ses applications geometriques.
       

140. Fourier biography
     
     • The first objection, made by Lagrange and Laplace in 1808, was to Fourier's expansions of functions as trigonometrical series, what we now call Fourier series.
       
     • Fourier's work provided the impetus for later work on trigonometric series and the theory of functions of a real variable.
       

141. Saks biography
     
     • One of the works for which Saks is most famous is their joint book Analytic functions which appeared in 1938 as volume eight in the Mathematical Monographs series.
       
     • This was not Saks' first monograph, however, for he had already published an important volume in the Mathematical Monographs series.
       
     • This earlier volume, the volume two in the series published in 1933, was his famous work Theory of the integral.
       

142. Schramm biography
     
     • His work in a spectacular series of papers has led to major progress in probability theory, in the theory of percolation and of random walks, as well as in related topics of conformal field theory.
       
     • He was also invited to give the prestigious Coxeter Lecture Series at the Fields Institute in September 2005.
       
     • He gave the following abstract for his three lecture series on Scaling limits of two dimensional random systems:- .
       

143. Puiseux biography
     
     • During this period he published a series of papers in Liouville's Journal.
       
     • He examined series expansions and looked at series with fractional powers.
       

144. Geiringer biography
     
     • This was awarded in 1917 for a thesis on Fourier series in two variables.
       
     • The debate over Geiringer's theses for Habilitation opens up a chapter of the history of mathematical statistics, namely, expansions of a discrete distribution with an infinite number of values in a series in successive derivatives of the Poisson distribution with respect to the parameter.
       
     • She wrote up her outstanding series of lectures on the geometrical foundations of mechanics and, although they were never properly published, these were widely used in the United States for many years.
       

145. Plemelj biography
     
     • He was soon making his own mathematical discoveries, for example he discovered for himself the series expansion for sin x and for cos x.
       
     • The way he did this was to first find the series expansion for arcsin x and then invert the series to obtain that for cos x.
       

146. Paley biography
     
     • Paley had already proved impressive results on Fourier series and had collaborated with Littlewood, his supervisor.
       
     • Zygmund's book Trigonometric Series published in 1935 owes a debt to the joint work that he carried out with Paley.
       
     • Soon after his arrival in America, however, certain studies of lacunary series which Paley had already begun suggested a new attack on the theory of interpolation and allied trigonometrical problems.
       

147. Rutishauser biography
     
     • Also in 1951 the first of a series of four papers by Rutishauser, Eduard Stiefel and Ambros Speiser, Programmgesteuerte digitale Rechengerate (elektronische Rechenmaschinen) appearing in 1950 and 1951.
       
     • In this series of papers the authors discuss in very considerable detail a number of the important mathematical questions that naturally arise in the design of a digital computer.
       
     • The QD algorithm represents a number of computational schemes for doing a surprising number of jobs: e.g., getting all eigenvalues of a matrix from its Schwarz constants, getting the zeros of a polynomial from its coefficients, finding the poles of a function from its power series, obtaining partial fraction representations of functions, and so forth.
       

148. Burkill biography
     
     • He resumed his research in mathematics winning the Adams prize in 1948 for an essay on integrals and trigonometric series.
       
     • The book covers: sets and functions, metric spaces, continuous functions on metric spaces, real and complex limits and series, uniform convergence, Riemann-Stieltjes integration, multivariable differential and integral calculus, Fourier series, Cauchy's theorem, Laurent expansions, residue calculus, infinite products, the factor theorem of Weierstrass, asymptotic expansions, and applications to special functions in particular the gamma function.
       

149. Pitt biography
     
     • He was tutored by J C Burkill and attended courses by world-leading mathematicians such as: functions of a complex variable from A E Ingham, almost periodic functions from A S Besicovitch, the theory of functions from J E Littlewood, and divergent series from G H Hardy.
       
     • Few research students can have had a more productive beginning to their careers for, after publishing A note on bilinear forms in 1936, and Theorems on Fourier series and powers series in 1937, he then published no fewer than eight papers in 1938.
       

150. Box biography
     
     • The main areas to which Box has contributed are: statistical inference, robustness, and modelling strategy; experimental design and response surface methodology; time series analysis and forecasting; distribution theory, transformation of variables, and nonlinear estimation; and applications of statistics.
       
     • Times series analysis.
       
     • It is basically a series of thirteen papers published by the authors and their co-authors, between 1962 and 1968, cobbled together with a minimum of re-writing.
       

151. Sargent biography
     
     • She did produce good results, despite any feelings that she may have had about them, for she published On Young's criteria for the convergence of Fourier series and their conjugates in the Proceedings of the Cambridge Philosophical Society based on this work which appeared in print in 1929.
       
     • Another paper published in the same year On the summability (C) of allied series and the existence of (CP) extends the conditions for the Cesaro summability of Fourier-Lebesgue series and of their conjugates given by Bosanquet in 1937 to the case of functions integrable in the Cesaro-Perron sense.
       

152. Runge biography
     
     • Runge then worked on a procedure for the numerical solution of algebraic equations in which the roots were expressed as infinite series of rational functions of the coefficients.
       
     • He succeeded in arranging the spectral lines of helium in two spectral series and, until 1897, this was thought to be evidence that hydrogen was a mixture of two elements.
       
     • In these spectra we found for the first time series systems of two different multiplicities.
       

153. Hall biography
     
     • In June 1939 Hall gave a series of lectures at a small meeting at the Mathematical Institute in Gottingen.
       
     • In August 1957 Hall gave a series of lectures at the Canadian Mathematical Congress Summer Seminar in Edmonton, Canada, on nilpotent groups which have had great influence ever since.
       
     • Besides containing a discussion of the possible order types of abelian series in simple groups, the paper also presents an extremely informative survey of the inter-relations that are known or conjectured to exist between the various classes of generalized soluble groups.
       

154. Steklov biography
     
     • He wrote General Theory of Fundamental Functions in which he examined expansions of functions as series in an infinite system of orthogonal eigenfunctions.
       
     • Steklov was not the first to examine series expansions in terms of infinite sets of orthogonal eigenfunctions, of course Fourier had examined a special case of this situation many years before.
       
     • He studied a generalisation of Parseval's equality for Fourier series to his general setting showing this to be a fundamental property.
       

155. Schafer biography
     
     • The curve is assumed to be analytic and thus can be represented in the neighbourhood of the point in question by power series.
       
     • Choice of the proper projective coordinate system permits the reduction of these power series to simple canonical forms.
       
     • With the aid of these canonical power series, [Schafer] derives numerous theorems concerning surfaces which osculate the given curve at the singular point, concerning the sections of the tangent developable in the neighbourhood of the singular point and concerning the projections of the given curve.
       

156. Smullyan biography
     
     • Smullyan's publications have been quite remarkable with the two outstanding books on retrograde analysis chess problems [The Chess Mysteries of the Sherlock Holmes (New York, 1979).',2)">2] and [The Chess Mysteries of the Arabian Knights (New York, 1981).',3)">3], a whole series of marvellous popular puzzle books such as [What is the name of this book? (New York, 1978).',1)">1] and [Satan, Cantor, and Infinity and other mind-boggling puzzles (New York, 1992).',4)">4], and some books on the foundations of mathematics and mathematical logic which are in many ways in a class of their own.
       
     • This book was the first of a series of texts which appeared in quick succession.
       
     • A third volume in the series Diagonalization and self-reference was published in 1994 and presents a very difficult topic in such a way as to make it both understandable and enjoyable.
       

157. Gelfond biography
     
     • (In 1966 Alan Baker proved Gelfond's Conjecture in general.) Gelfond's papers in 1933 and 1934, which include his remarkable achievement, are: Gram determinants for stationary series (written jointly with Khinchin) (1933); A necessary and sufficient criterion for the transcendence of a number (1933); Functions that take integer values at the points of a geometric progression (1933); On the seventh problem of D Hilbert (1934); and On the seventh problem of Hilbert (1934).
       
     • The chapter titles of this book are: Residues; Singular points and series representations of a function; Expansion of a function in a series and properties of the gamma function; Some functional identities and asymptotic estimates; and Laplace transformation and some problems which are solved by the use of residue theory.
       

158. Kaprekar biography
     
     • The possible digitadition series are separated into three types: type A has all is members coprime to 3; type B has all is members divisible by 3 but not by 9; C has all is members divisible by 9.
       
     • Kaprekar notes that if x and y are of the same type (that is, each prime to 3, or each divisible by 3 but not 9, or each divisible by 9) then their digitadition series coincide after a certain point.
       
     • He conjectured that a digitadition series cannot contain more than 4 consecutive primes.
       

159. Francesca biography
     
     • Piero almost certainly wrote all three works in the vernacular (his native dialect was Tuscan), and all three are in the style associated with the tradition of 'practical mathematics', that is, they consist largely of series of worked examples, with rather little discursive text.
       
     • It deals with arithmetic, starting with the use of fractions, and works through series of standard problems, then it turns to algebra, and works through similarly standard problems, then it turns to geometry and works through rather more problems than is standard before (without warning) coming up with some entirely original three-dimensional problems involving two of the 'Archimedean polyhedra' (those now known as the truncated tetrahedron and the cuboctahedron).
       
     • He accordingly starts with a series of mathematical theorems, some taken from the optical work of Euclid (possibly through medieval sources) but some original to Piero himself.
       

160. Fubini biography
     
     • A series of decrees removed Jews from positions of influence in government, banking and education.
       
     • His technical mastery often permitted him to discover simpler demonstrations of such theorems as those of Berstein and Pringsheim on the development of Taylor series.
       
     • This led him to solve a whole series of engineering problems which he was writing up as a textbook towards the end of his life.
       

161. Skopin biography
     
     • The first significant result which Skopin produced was concerned with the upper central series of groups [Doklady Akad.
       
     • From the 1970's, he became interested in problems concerning the structure of the lower central series of groups of the Burnside type, that is groups of prime-power exponent.
       
     • It was very natural to fix the exponent and the number of generators and to study the lower central series of a group in more detail by performing direct calculations of its factors.
       

162. Tukey biography
     
     • Tukey's first major contribution to statistics was his introduction of modern techniques for the estimation of spectra of time series.
       
     • And when I have pondered about why such techniques as the spectrum analysis of time series have proved so useful, it has become clear that their 'dealing with fluctuations' aspects are, in many circumstances, of lesser importance than the aspects that would already have been required to deal effectively with the simpler case of very extensive data where fluctuations would no longer be a problem.
       
     • Time series : 1949-1964 (Belmont, CA, 1984).',2)">2]:- .
       

163. Jones biography
     
     • It included the differential calculus, infinite series, and is also famed since the symbol π is used in it with its modern meaning.
       
     • These included transcripts of Newton's manuscripts, letters and results obtained with the method of infinite series which Newton had discovered in about 1664.
       
     • With assistance from Newton himself, Jones produced Analysis per quantitatum series, fluxiones, ac differentia in 1711 although it should be noted that this first edition of 1711 did not record either Newton's name nor that of Jones.
       

164. Langlands biography
     
     • Then, over the next couple of years, he produced deep results on Eisenstein series and went on to apply Eisenstein series to prove a number theory conjecture due to Weil.
       
     • He received the Cole Prize in Number Theory from the American Mathematical Society in 1982 for his pioneering work on automorphic forms Eisenstein series, and product formulae.
       

165. Rota biography
     
     • As we have indicated above, Rota worked on functional analysis for his doctorate and, up to about 1960, he wrote a series of papers on operator theory.
       
     • This paper was the first of a series of ten papers with this main title, all ten have subtitles (for example this first one was subtitled Theory of Mobius functions ) and all the remaining nine have between one and three additional co-authors.
       
     • He had been due to give a series of three lectures at Temple University, the Groswald Memorial Lectures, on the previous day and, when he failed to arrive in Philadelphia, a check was made at his home.
       

166. Yule biography
     
     • In 1895 Yule was elected to the Royal Statistical Society and over the next few years, inspired by Pearson, he produced a series of important articles on the statistics of regression and correlation.
       
     • He wrote papers on time-correlation in which he introduced the correlogram and he did fundamental work on the theory of autoregressive series.
       
     • The last chapters discuss interpolation and graduation, index numbers, and time series.
       

167. Balmer biography
     
     • However, despite being a mathematics teacher and lecturer all his life, Balmer is best remembered for his work on spectral series and his formula, given in 1885, for the wavelengths of the spectral lines of the hydrogen atom.
       
     • In his paper of 1885 Balmer suggested that giving n other small integer values would give the wavelengths of other series produced by the hydrogen atom.
       
     • Indeed this prediction turned out to be correct and these series of lines were later observed.
       

168. Genocchi biography
     
     • After a series of defeats, Charles Albert's army withdrew from Milan.
       
     • The main research topics which Genocchi worked on were number theory, series and the integral calculus.
       

169. Newton biography
     
     • Newton made notes on Wallis's treatment of series but also devised his own proofs of the theorems writing:- .
       
     • The title page of Analysis per quantitatum series, fluxiones (1711) .
       

170. Householder biography
     
     • In a remarkable series of papers he effectively classified the algorithms for solving linear equations and computing eigensystems, showing that in many cases essentially the same algorithm had been presented in a large variety of superficially quite different algorithms.
       
     • At the 13th in the series of Householder Symposia held in Pontresina, Switzerland in 1996, Friedrich L Bauer spoke on Memories of Alston Householder.
       

171. Motzkin biography
     
     • Motzkin's first publication, however, was not on linear programming but rather on power series.
       
     • Both linear programming and power series were themes which ran through Motzkin's research throughout his life but he was an extremely broad mathematician and there were many other themes.
       

172. Jung biography
     
     • The analytic approach, followed in the present book, uses as fundamental concept that of a place of the field K, a place being an isomorphism of K into the quotient field of the ring of convergent power series in two variables (the uniformizing variables at the place) with the requirement that distinct couples of values of the variables u, v in these power series sufficiently near to (0, 0) should lead to distinct values for some function of the field.
       

173. Frobenius biography
     
     • On the development of analytic functions in series.
       
     • In a series of letters to Dedekind, the first on 12 April 1896, his ideas on group characters quickly developed.
       

174. Plessner biography
     
     • In 1921 Plessner went to Gottingen where he took courses on Dirichlet series and Galois theory by Edmund Landau; algebraic number fields by Emmy Noether; and calculus of variations by Courant.
       
     • Plessner obtained his doctorate from Giessen in 1922 for a thesis on conjugate trigonometrical series.
       

175. Plancherel biography
     
     • In a series of articles he generalized results in the classical Fourier theory to more general spaces (Hilbert spaces) by investigating various orthonormal systems of functions, their summability and the representation of functions in such systems by Fourier series and Fourier integrals and more general integral transformations.
       

176. Lefschetz biography
     
     • During these years he wrote a series of important papers on topology despite being out the mainstream of mathematical research.
       
     • Lefschetz had many students working in this area and, between 1950 and 1960, a series of important publications Contributions to the theory of nonlinear oscillations appeared in the Annals of Mathematics Studies, published by Princeton University Press.
       

177. De Finetti biography
     
     • In this regard, D V Lindley [Journal of the Royal Statistical Society, Series A 149 (1986), 252.',6)">6], [Encyclopedia of Statistical Sciences (Supplement) (New York, 1989), 46-47.',7)">7] reports that Bruno de Finetti was especially fond of the aphorism:- .
       
     • A key tool for him was nomenclature: for example, as reported by D V Lindley [Journal of the Royal Statistical Society, Series A 149 (1986), 252.',6)">6], [Encyclopedia of Statistical Sciences (Supplement) (New York, 1989), 46-47.',7)">7], he insisted that "random variables" should more appropriately be called "random quantities", for "What varies?".
       

178. Seifert biography
     
     • The book was accepted by Blaschke for the Hamburg monograph series but the two authors ran into problems with a Latin epigraph which they wished to put at the beginning.
       
     • Seifert, still able to do mathematical research, worked on differential equations and wrote a series of papers on the topic through the war years.
       

179. Gibbs biography
     
     • A series of five papers by Gibbs on the electromagnetic theory of light were published between 1882 and 1889.
       
     • The American Mathematical Society named a lecture series in honour of Gibbs.
       

180. Whiteside biography
     
     • An abundance of references direct the reader either to the contemporary literature or to the previous volumes of this series where related problems have already been treated.
       
     • This, however, did not end up in a series of major publications as his work on Newton had, rather he published only a few scraps.
       

181. Selberg biography
     
     • automorphic functions, Dirichlet series.
       
     • Secondly, his papers up to 1947, which appeared mostly in Norwegian series or journals of limited distribution and partly even during World War II, are now at last easily accessible.
       

182. Airey biography
     
     • London, Series A, 94 (661) (1918), 307-314.
       
     • London, Series A, 96 (674) (1919), 1-8.
       

183. Munn biography
     
     • A series of papers between 1966 and 1973 exploring these ideas gave rise to results that are now regarded as classical.
       
     • His discovery of Passman's books on infinite group rings brought about a further change in the main thrust of his work, and in the eighties, while still writing the occasional paper on pure semigroup theory, he returned to the study of semigroup algebras, publishing a series of remarkable papers linking semigroup properties to ring-theoretic properties of their algebras.
       

184. Loewner biography
     
     • He wrote a series of papers on this topic, culminating in one where he proved a special case of the Bieberbach conjecture in 1923.
       
     • given by the series .
       

185. Appell biography
     
     • In 1880 Appell defined a series of functions satisfying the condition that the derivative of the nth function is n times the (n - 1)th function.
       
     • his scientific work consists of a series of brilliant solutions of particular problems, some of the greatest difficulty.
       

186. Peter biography
     
     • He suggested Peter examine Godel's work and in a series of papers she became a founder of recursive function theory.
       
     • In a series of articles, beginning in 1934, Peter developed various deep theorems about primitive recursive functions, most of them with an explicit algorithmic content.
       

187. Petersson biography
     
     • In 1982 Petersson published an important book Modulfunktionen und quadratische Formen in Springer-Verlag's Ergebnisse der Mathematik und ihrer Grenzgebiete series.
       
     • Petersson continued to undertake important research after he retired in 1970 and his final paper Uber Spuren von Modulformen und die Eisensteinschen Reihen in den Kongruenzklassen der rationalen Modulgruppe (On traces of modular forms and the Eisenstein series in the congruence classes of the rational modular group) appeared in 1986, two years after his death.
       

188. Riesz biography
     
     • A satisfactory theory of series of orthonormal functions only became possible after the invention of the Lebesgue integral and this theory was largely the work of Riesz.
       
     • He also studied orthonormal series and topology.
       

189. Rogers James biography
     
     • Such names as Rogers-Ramanujan identities, Rogers-Ramanujan continued fractions and Rogers transformations are known in the theory of partitions, combinatorics and hypergeometric series.
       
     • [Ramanujan (New York, 1940).',2)">2], [q-series: their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, (Providence, 1986).',4)">4], [Math.
       

190. Bernoulli Johann biography
     
     • In 1694 he considered the function y = xx and he also investigated series using the method of integration by parts.
       
     • He summed series, and discovered addition theorems for trigonometric and hyperbolic functions using the differential equations they satisfy.
       

191. Subbotin biography
     
     • Subbotin not only showed the possibility of improving the convergence of the trigonometric series by which the behaviour of perturbing forces is represented, but also gave an expression for determining Laplace coefficients and presented formulas for computing the coefficients of the necessary members of the trigonometric series.
       

192. Cherry biography
     
     • Since the author uses the solutions of Chaplygin, in the form of an infinite series of hypergeometric functions, of the linear second order partial differential equation in the hodograph variables of the potential function, this series diverges for values of the velocity whose speeds exceed the speed at infinity.
       

193. Russell Scott biography
     
     • These reports, in fact all Russell's own work, contain a remarkable series of observations, at sea, in rivers and canals, and in Russell's own wave tank constructed for the purpose.
       
     • In M Lakshmanan, Solitons, Springer Series in Nonlinear Dynamics, (New York, 1988) 150-281.',4)">4] and [Acta Applicandae Mathematicae 39 (1995), 193-228.',5)">5].
       

194. Linnik biography
     
     • From this he was able to produce a whole series of papers proving powerful arithmetical consequences, including a variant of the Goldbach Conjecture.
       
     • Also in 1967 Linnik published Lecons sur les problemes de statistique analytique which came about as the result of a series of lectures he had given in the previous year at the Institut de Statistique of the University of Paris.
       

195. Rutherford biography
     
     • Rutherford wrote several other texts in the Oliver and Boyd series which he set up with Aitken.
       
     • Outstanding research contributions led to Rutherford being elected a fellow of the Royal Society of Edinburgh in 1934 and he received the Keith Prize from the Society for an outstanding series of papers he published in 1951-53.
       

196. Aitken biography
     
     • With Rutherford he was editor of a series of the University Mathematical Texts and he himself wrote for the series Determinants and matrices (1939) and Statistical Mathematics (1939).
       

197. Frenet biography
     
     • He published these observations in the Memoires de l'Academie imperiale de Lyon in a continuing series, first in 1853, next in 1856, and again in 1858.
       
     • A Drian took over publishing the series of meteorological observations.
       

198. Littlewood biography
     
     • For 35 years he collaborated with G H Hardy working on the theory of series, the Riemann zeta function, inequalities, and the theory of functions.
       
     • The collaboration led to a series of papers Partitio numerorum using the Hardy-Littlewood-Ramanujan analytical method.
       

199. Ford biography
     
     • In addition to his work on point-wise discontinuous functions which we mentioned above, Ford is best known for an "absolutely marvellous geometric interpretation of the Farey series".
       
     • Still others have to do with Ford's former activity as editor of the MONTHLY, where he started the series of papers with the titles "What is ..
       

200. Mercator Gerardus biography
     
     • His 'atlas' continued with a further series of maps of France, Germany and the Netherlands in 1585.
       
     • Although the project was never completed Mercator did publish a further series in 1589 including maps to the Balkans (then called Sclavonia) and Greece.
       

201. Pringsheim biography
     
     • He also has important results on the singularities of power series with positive coefficients.
       
     • In 1893 he proved that a function is analytic if it is infinitely differentiable on an open interval and the radius of convergence r(x) of the Taylor series centred at x is bounded away from 0.
       

202. Osgood biography
     
     • Some papers over the next few years included: Sufficient conditions in the calculus of variations (1900), On a fundamental property of a minimum in the calculus of variations and the proof of a theorem of Weierstrass's (1901), A Jordan curve of positive area (1903), On Cantor's theorem concerning the coefficients of a convergent trigonometric series, with generalizations (1909), On the gyroscope (1922), and On normal forms of differential equations (1925).
       
     • Other classic texts included Introduction to Infinite Series (1897), A First Course in the Differential and Integral Calculus (1909), Topics in the theory of functions of several complex variables published by the American Mathematical Society in 1914, Plane and Solid Analytic Geometry (with W C Graustein, 1921), Advanced Calculus (1925), and Mechanics (1937).
       

203. Young biography
     
     • He studied Fourier series and orthogonal series in general, the ideas which he put forward being further developed by Littlewood and Hardy.
       

204. Bass biography
     
     • Kaplansky gave an inspiring series of courses on homological methods in commutative algebra.
       
     • Bass produced a series of papers during his first years at Columbia, for example Finitistic dimension and a homological generalization of semiprimary rings (1960), Projective modules over algebras (1961), Injective dimension in noetherian rings (1962), and Torsion free and projective modules (1962).
       

205. Hedrick biography
     
     • He was editor of the American Mathematical Monthly from 1913 to 1915, editor-in-chief of the Bulletin of the American Mathematical Society from 1921 to 1937, he was editor of 34 volumes in the Engineering Science Series and 35 volumes in the Series of Mathematical Texts.
       

206. Chernikov biography
     
     • In Chernikov's articles, therefore, a series of geometrically obvious properties of linear inequalities is given in analytic form that is more convenient for the use of machine techniques.
       
     • A series of papers by Chernikov in the 1960s studied polyhedrally closed systems, special types of infinite systems of linear inequalities [Uspekhi Mat.
       

207. Siegel biography
     
     • In this general area Siegel considered the theory of discontinuous groups and their fundamental domains, algebraic relations between modular functions and between modular forms, and Fourier series of modular forms.
       
     • He examined Birkhoff's work on perturbation theory solutions for analytical Hamiltonian differential equations near an equilibrium point using formal power series.
       

208. Salmon biography
     
     • The first of these was his sermon Prayer published in 1849, which was followed by a series of publications of his sermons, for example many are collected in Sermons preached in Trinity College Chapel (1861) and Cathedral and University Sermons (1900).
       
     • He published much in the area of theology with works such as The eternity of future punishment (1864), The reign of law (1873), Non-miraculous Christianity (1881), Introduction to the New Testament (1885), The infallibility of the Church (1888), Thoughts on the textual criticism of the New Testament (1897), and a series of articles between 1877 and 1887 on the history of the early Christian Church in the Dictionary of Christian Biography.
       

209. Spottiswoode biography
     
     • His series of memoirs on the contact of curves and surfaces, contributed to the 'Philosophical Transactions' of 1862 and subsequent years, mainly gave him his high rank as a mathematician.
       
     • The interesting series of communications on the contact of curves and surfaces which are contained in the Philosophical Transactions of 1862 and subsequent years would alone account for the high rank he obtained as a mathematician.
       

210. Hille biography
     
     • Hille's main work was on integral equations, differential equations, special functions, Dirichlet series and Fourier series.
       

211. Gateaux biography
     
     • Volterra himself, invited by Borel and Hadamard, came to Paris to give a series of lectures on functional analysis, published in 1913 ([Lecons sur les fonctions de lignes (Gauthier-Villars, Paris, 1913).',23)">23]) and whose redaction was precisely made by Peres.
       
     • Nobody could predict the development and the range that this new series of researches could have attained.
       

212. Kingman biography
     
     • He wrote on the theory of Markov processes and published an important series of articles on Markov transition probabilities which we gave details of above.
       
     • In 1979 Kingman gave a series of lectures at Iowa State University on the contributions of mathematics to the study of genetic evolution.
       

213. Bouquet biography
     
     • With Briot he worked from 1853 onwards on deep studies of Cauchy's ideas of analysis and produced many fundamental papers on series expansions of functions and on elliptic functions.
       
     • In 1853 they established conditions for a function to be expandable into an entire series in their important paper Note sur le developpement des fonctions en series convergentes, ordonnees suivant les puissances croissantes de la variable.
       

214. Schlomilch biography
     
     • In 1847 he gave a general remainder formula for the remainder in Taylor series.
       
     • He discovered an important series expansion of an arbitrary function in terms of Bessel functions in 1857.
       
     • In a militant manner, Barfuss defended an obsolete point of view on the question of symbolic calculation with divergent series against Schlomilch.
       

215. Puri biography
     
     • Time series and related topics.
       
     • Professor Puri's research has lead to his being considered one of the most versatile and prolific researchers in the world in mathematical statistics in the areas of nonparametric statistics, order statistics, limit theory under mixing, time series, splines, tests of normality, generalized inverses of matrices and related topics, stochastic processes, statistics of directional data, random sets, and fuzzy sets and fuzzy measures.
       

216. Cartan Henri biography
     
     • Cartan published Les transformations analytiques des domaines cercles les uns dans les autres in 1930 and, since this paper contained generalisations of results proved by Heinrich Behnke, he was invited by Behnke to visit Germany in May 1931 and give a series of lectures at Munster in Westphalen where Behnke taught.
       
     • A central figure in this development has been Henri Cartan, whose series of papers in this field starting in the 1920's dealt with fundamental questions relating to Nevanlinna theory, generalizations of the Mittag-Leffler and Weierstrass theorems to functions of several variables, problems concerned with biholomorphic mappings and the biholomorphic equivalence problem, domains of holomorphy and holomorphic convexity, etc.
       

217. Fatou biography
     
     • We should also mention his work on Taylor series where he examined the convergence and the analytic extension of the series.
       

218. Fibonacci biography
     
     • There are also problems involving perfect numbers, problems involving the Chinese remainder theorem and problems involving summing arithmetic and geometric series.
       
     • For unity is a square and from it is produced the first square, namely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers.
       

219. Bugaev biography
     
     • He wrote a Master's thesis on convergence of infinite series at the University of Moscow which he submitted and successfully defended in 1863.
       
     • (6) (1963), 71-78.',8)">8] is concerned with the subsequent development of ideas on general tests of convergence of infinite series contained in Bugaev's thesis.
       

220. Cassini biography
     
     • With these instruments Cassini made a series of new discoveries.
       
     • He published detailed series of observations of the moons of Jupiter in 1668.
       

221. Zorn biography
     
     • Or we may prescribe a seemingly much more powerful condition, namely, that the function possesses a development into (abstract) power series about each point of the domain of definition.
       
     • For it turns out that only a very weak continuity property has to be added to the existence of the Gateaux differential in order to ensure the existence of the power series development called for by the second definition.
       

222. Stormer biography
     
     • It is unusual for anyone to publish a paper in the year they enter university as an undergraduate, but this is exactly what Stormer did with a paper on the summation of trigonometric series.
       
     • His output of mathematical papers continued with twelve papers on series, number theory, and the theory of functions between 1896 and 1902.
       

223. Henrici Peter biography
     
     • The first paper he published in English was A Neumann series for the product of two Whittaker functions which appeared in the Proceedings of the American Mathematical Society in 1953.
       
     • The first volume Power series - integration - conformal mapping - location of zeros first appeared in 1974, the second volume Special functions - integral transforms - asymptotics - continued fractions first appeared in 1977, and the third and final volume Discrete Fourier analysis - Cauchy integrals - construction of conformal maps - univalent functions first appeared in 1986.
       

224. Tutte biography
     
     • As a young mathematician and codebreaker, he deciphered a series of German military encryption codes known as FISH.
       
     • Among his books are: Connectivity in graphs published in 1966; Introduction to the theory of matroids (1971), based on a series of lectures given by Tutte at the Rand Corporation in 1965; Graph Theory (1984); and Graph Theory as I Have Known It (1998) which gives a fascinating account of how he discovered his many fundamental results.
       

225. Schrodinger biography
     
     • Schrodinger published his revolutionary work relating to wave mechanics and the general theory of relativity in a series of six papers in 1926.
       
     • After giving a brilliant series of lectures in Madison he was offered a permanent professorship there but [Schrodinger : Life and Thought (New York, 1989).',8)">8]:- .
       

226. Faber biography
     
     • In 1902 he received a doctorate from the Ludwig-Maximilians University in Munich for a thesis Uber Reihenentwicklungen analytischer Funktionen (On the series expansion of analytical functions).
       
     • He received his university teaching qualification from the University of Wurzburg in 1905, after submitting his Habilitationsschrift presenting work on power series in several variables.
       

227. Montgomery biography
     
     • His interests turned from point-set topology to transformation groups quite early in his career and he published a series of papers on the topic in collaboration with Leo Zippin.
       
     • In a long series of papers written in the late 1960s and early 1970s, [Montgomery and C T Yang] used the study of group actions on homotopy 7-spheres to showcase and test the growing new techniques of differential topology, especially index theory and surgery theory.
       

228. Arbogast biography
     
     • Essentially he realised that there was no rigorous methods to deal with the convergence of series.
       
     • The formal algebraic manipulation of series investigated by Lagrange and Laplace in the 1770s was put in the form of operator equalities by Arbogast in 1800 in Calcul des derivations.
       

229. Kumano-Go biography
     
     • During these years Kumano-Go published a series of papers which studied the local and global uniqueness of the solutions of the Cauchy problem for partial differential equations.
       
     • In addition to his work on pseudo-differential operators, Kumano-Go published a series of papers on the product of Fourier integral operators.
       

230. Legendre biography
     
     • He then introduced what we call today the Legendre functions and used these to determine, using power series, the attraction of an ellipsoid at any exterior point.
       
     • The 1785 paper on number theory contains a number of important results such as the law of quadratic reciprocity for residues and the results that every arithmetic series with the first term coprime to the common difference contains an infinite number of primes.
       

231. Kuratowski biography
     
     • He was also one of the founders and an editor of the important Mathematical Monographs series.
       
     • He contributed the third volume in this series with his monograph on topology which we will mention again below.
       

232. Mellin biography
     
     • He applied this technique systematically in a long series of papers to the study of the gamma function, hypergeometric functions, Dirichlet series, the Riemann zeta function and related number-theoretic functions.
       

233. Goldstine biography
     
     • They produced a series of reports on the EDVAC (Electronic Differential Variable Computer) which changed the whole concept of computers.
       
     • He did not undertake research solely on computers and their applications, however, for he published a series of three papers on Hilbert space with non-associative scalars (1962, 1964, 1966).
       

234. Granville biography
     
     • She wrote a doctoral thesis On Laguerre Series in the Complex Domain and in 1949, together with Marjorie Lee Browne who graduated from the University of Michigan in the same year, she became the one of the first black American women to be awarded a Ph.D.
       
     • math must not be taught as a series of disconnected, meaningless technical procedures from dull and empty textbooks.
       

235. Schoenberg biography
     
     • Schoenberg made further outstanding contributions in a series of papers between 1950 and 1959 on the theory of Polya frequency functions.
       
     • He investigated their wide applications in approximation theory in a series of three papers between 1969 and 1973.
       

236. Lang biography
     
     • Lang's mathematical research ranged over a wide range of topics such as algebraic geometry, Diophantine geometry (a term Lang invented), transcendental number theory, Diophantine approximation, analytic number theory and its connections to representation theory, modular curves and their applications in number theory, L-series, hyperbolic geometry, Arakelov theory, and differential geometry.
       
     • Three public dialogues (1985), Introduction to complex hyperbolic spaces (1987), Introduction to Arakelov theory (1988), Topics in Nevanlinna theory (1990), Basic analysis of regularized series and products (1993), Fundamentals of differential geometry (1999), and Math talks for undergraduates (1999).
       

237. Bessel biography
     
     • His meeting with important English scientists, including Herschel, impressed him deeply and stimulated him to finish and publish, despite his weakened health, a series of works.
       
     • Bessel functions appear as coefficients in the series expansion of the indirect perturbation of a planet, that is the motion caused by the motion of the Sun caused by the perturbing body.
       

238. Peschl biography
     
     • The titles of the chapters are: Algebra and geometry of complex numbers; Fundamental topological concepts, sets, sequences of complex numbers and infinite series; Functions, real and complex differentiability and holomorphy; Integral theorems and their consequences; Winding number and curves homologous to zero; Taylor development of holomorphic functions; Elementary transcendental functions; Laurent series, isolated singularities and residue calculus; Holomorphic and meromorphic functions obtained by limiting processes; Analytic continuation; and Conformal mappings.
       

239. Book biography
     
     • He was editor of three monograph series, in particular being editor-in-chief of the series Progress in Theoretical Computer Science.
       

240. Rayleigh biography
     
     • Turning my attention to nitrogen, I made a series of determinations ..
       
     • Having obtained a series of concordant observations on gas thus prepared I was at first disposed to consider the work on nitrogen as finished.
       

241. Carslaw biography
     
     • an enthusiast in original research, and having studied the mathematical papers and memoirs bearing on Fourier's series and their application in mathematical physics, purposes writing a book on the subject.
       
     • The second book was Introduction to the theory of Fourier's series and integrals and the mathematical theory of the conduction of heat.
       

242. Eisenstein biography
     
     • developed his own independent analytic theory of elliptic functions, based on the technique of summing certain conditionally convergent series.
       
     • Moreover, this case provides not merely an illuminating introduction to his theory, but also the simplest proofs for a series of results, originally discussed by Euler ..
       

243. Titchmarsh biography
     
     • He studied Fourier series and Fourier integrals writing Introduction to the Theory of Fourier Integrals (1937).
       
     • From 1939 Titchmarsh concentrated on the theory of series expansions of eigenfunctions of differential equations, work which helped to resolve problems in quantum mechanics.
       

244. Speiser biography
     
     • There followed a series of four papers by Stiefel, Rutishauser and Speiser, Programmgesteuerte digitale Rechengerate (elektronische Rechenmaschinen) appearing in 1950 and 1951.
       
     • In this series of papers the authors discuss in very considerable detail a number of the important mathematical questions that naturally arise in the design of a digital computer.
       

245. Petersen biography
     
     • He wrote a series of school and undergraduate texts which achieved international acclaim despite being too difficult for all but the ablest pupils.
       
     • He wrote a series of textbooks based on courses he had given at the College of Technology: one on plane geometry in 1877; one on statics in 1881; one on kinematics in 1884; and one on dynamics in 1887.
       

246. Lerch biography
     
     • In that topic he studied infinite series, and the gamma function as well as other special functions.
       
     • He also studied the principle of most rapid convergence of a series.
       

247. Kerekjarto biography
     
     • The first of these courses was enlarged into a book Vorlesungen uber Topologie (Lectures on Topology) which appeared in the series Grundlehren der Mathematischen Wissenschaften in 1923.
       
     • His final work was intended to be a series of five books, only two of which were written before his death.
       

248. Brioschi biography
     
     • Brioschi used the findings of a series of major projects or participated in the projects' development - for example, in the regulation of the Po and Tiber ..
       
     • He made fundamental contributions in 1857 to changing the Annali di Scienze Matematiche e Fisiche into a journal of international standing, in 1886 to the new series of the journal Politecnico, to an Italian edition of Euclid's Elements for secondary education, and he edited Leonardo da Vinci's Codice Atlantico which was a major contribution to the understanding of the history of science and technology.
       

249. Herschel biography
     
     • He studied algebras and published papers on trigonometrical series.
       
     • I am going under my father's directions, to take up the series of his observations where he has left them (for he has now pretty well given over regular observing) and continuing his scrutiny of the heavens with powerful telescopes ..
       

250. Artin biography
     
     • He defined a new type of L-series, which generalised Dirichlet's L-series, yet was quite different in nature.
       

251. Gelfand biography
     
     • He saw the importance of the work of Sobolev and Schwartz on the theory of generalised functions and distributions, and he developed this theory in a series of monographs.
       
     • Between 1968 and 1972 Gelfand produced a series of important papers on the cohomology of infinite dimensional Lie algebras.
       

252. Ferrar biography
     
     • Ferrar wrote many research papers which deal with the convergence of series, an interest which came from working with G N Watson at Cambridge for during a summer vacation while an undergraduate.
       
     • From about 1930 his interests turned towards number theory and he examined the convergence of series and the evaluation of singular integrals.
       

253. Hecke biography
     
     • Schoeneberg describes Hecke's contributions to a number of topics which he lists as follows: Hilbert modular functions, Dedekind zeta functions, arithmetical notions and methods, elliptic modular forms of level N, algebraic functions, Dirichlet series with functional equation, Hecke-operators Tn, and physics where he made contributions to the kinetic theory of gases.
       
     • He then introduced the new concept of "Grossencharakter" and the corresponding L-series, to which he extended the properties of analytic continuation he had proved for the zeta functions in 1917.
       

254. Almgren biography
     
     • The reasons for this are partly indicated, for readers with only an advanced calculus background, in terms of examples, illustrated by a series of rather beautiful diagrams in colour.
       
     • Throughout his career, Almgren brought great geometric insight, technical power, and relentless determination to bear on a series of the most important and difficult problems in his field.
       

255. Krawtchouk biography
     
     • Later in 1918 an independent Ukraine was again declared in Kiev but there followed a series of struggles between Ukrainian nationalist, White, and Red forces.
       
     • There was a series of public trials and in a massive terror campaign against the population as a whole.
       

256. Babbage biography
     
     • He wrote papers on several different mathematical topics over the next few years but none are particularly important and some, such as his work on infinite series, are clearly incorrect.
       
     • The drawings of the Analytical Engine have been made entirely at my own cost: I instituted a long series of experiments for the purpose of reducing the expense of its construction to limits which might be within the means I could myself afford to supply.
       

257. Hadamard biography
     
     • Already at this stage he began to undertake research, investigating the problem of finding an estimate for the determinant generated by coefficients of a power series.
       
     • Hadamard obtained his doctorate in 1892 for a thesis on functions defined by Taylor series.
       

258. Quetelet biography
     
     • Chance, that mysterious, much abused word, should be considered only a veil for our ignorance; it is a phantom which exercises the most absolute empire over the common mind, accustomed to consider events only as isolated, but which is reduced to naught before the philosopher, whose eye embraces a long series of events and whose penetration is not led astray by variations, which disappear when he gives himself sufficient perspective to seize the laws of nature.
       
     • It seems to me that that which relates to the human species, considered en masse, is of the order of physical facts: the greater the number of individuals, the more the influence of the individual will is effaced, being replaced by the series of general facts that depend on the general causes according to which society exists and maintains itself.
       

259. McCrea biography
     
     • He proved conclusively that this was right in a paper which gave perhaps the first qualitative correct model of the solar atmosphere, with the currently accepted abundance of about three-quarters hydrogen and one-quarter helium by mass, and led to a series of equally authoritative papers on the atmospheres of other stars.
       
     • This was in the Oliver and Boyd series and McCrea writes in the Preface:- .
       

260. Shewhart biography
     
     • In this classic volume, based on a series of ground-breaking lectures given to the Graduate School of the Department of Agriculture in 1938, Dr Shewhart illuminates the fundamental principles and techniques basic to the efficient use of statistical method in attaining statistical control, establishing tolerance limits, presenting data, and specifying accuracy and precision.
       
     • During 1944-46 he served on the National Research Council and for over 20 years he served as editor of the Mathematical Statistics Series of John Wiley and Sons.
       

261. Sneddon biography
     
     • The first was Special functions of mathematical physics and chemistry published in the Oliver and Boyd series in 1956.
       
     • It was, as all the Oliver and Boyd series books, sold at a price a student could afford and it provided a straightforward account of the topic in a short but very clear style.
       

262. Stieltjes biography
     
     • He received his doctorate of science in 1886 for a thesis on asymptotic series.
       
     • Also important is his work on divergent series and discontinuous functions.
       

263. Barnes biography
     
     • Barnes' episcopate was marked by a series of controversies stemming from his outspoken views and, rather surprisingly for someone who held such high office in the Church, often unorthodox religious beliefs.
       
     • Barnes next turned his attention to the theory of integral functions, where, in a series of papers, he investigated their asymptotic structure.
       

264. Tikhonov biography
     
     • After a series of fundamental papers introducing the topic, the work was carried on by his students.
       
     • In the 1960s Tikhonov began to produce an important series of papers on ill-posed problems.
       

265. Chernoff biography
     
     • This was in fact Chernoff's third paper since he published A note on the inversion of power series in 1947 which:- .
       
     • [treats] the multiplication of power series and their inversion by means of a movable strip of paper on which the coefficients are written.
       

266. Pincherle biography
     
     • Remaining faithful to the ideas of Weierstrass, he did not take the topological approach that later proved to be most successful, but tried to start from a series of powers of the D derivation symbol.
       
     • Although his efforts did not prove very fruitful, he was able to study in depth the Laplace transform, iteration problems, and series of generalised factors.
       

267. Albert Abraham biography
     
     • In addition he was beset by a series of illnesses ..
       
     • These matrices arise in the theory of complex manifolds and Albert went on to write an important series of papers on these questions over the following years.
       

268. Kruskal William biography
     
     • In 1954, prior to the era of modern high speed computers, the present authors published the first of a series of four landmark papers on measures of association for cross classifications.
       
     • This series of papers evolved over a twenty-year period.
       

269. Knopp biography
     
     • Chapter III: Sets, sequences and power series.
       
     • Volume 1 covers numbers, functions, limits, analytic geometry, algebra, set theory; volume 2 covers differential calculus, infinite series, elements of differential geometry and of function theory; and volume 3 covers integral calculus and its applications, function theory, differential equations.
       

270. Wald biography
     
     • During this period Wald published 10 papers on economics and econometrics, and he also published an important monograph in 1936 on seasonal movements in time series.
       
     • seasonal corrections to time series, approximate formulas for economic index numbers, indifference surfaces, the existence and uniqueness of solutions of extended forms of the Walrasian system of equations of production, the Cournot duopoly problem, and finally, in his much used work written with Mann (1943), stochastic difference equations.
       

271. Sasaki biography
     
     • One of them is a series of three papers on the relations between the structure of spaces with normal conformal connections and their holonomy groups.
       
     • It was this last series of three papers which formed the basis of Sasaki's doctoral thesis which he presented in 1943, receiving his doctorate in July of that year.
       

272. Hammer biography
     
     • There followed a series of papers, typical of which are Linear Programming and Transportation (1960), Applications of Mathematics to Economics (1960), Optimization of the Development Plan of an Industry (1961), and A Method for Solving Transportation Problems (1961).
       
     • RUTCOR played a major role, with Hammer as its director, with regular series of seminars, workshops, and courses put on for the numerous graduate students.
       

273. Fasenmyer biography
     
     • There her doctoral studies were supervised by Earl Rainville who suggest that she examine some combinatorial problems related to hypergeometric series.
       
     • In her thesis she gave algorithms to find recurrence relations between sums of terms in hypergeometric series.
       

274. Schlesinger biography
     
     • In 1926 Schlesinger published a book on Lebesgue integration and Fourier series in collaboration with Abraham Plessner.
       
     • The work studies trigonometric series and the boundary behaviour of analytic functions.
       

275. Brouncker biography
     
     • The English Civil War broke out in 1642, the Scots joined the Parliament forces and Charles I suffered a series of defeats.
       
     • Brouncker's mathematical achievements includes work on continued fractions and calculating logarithms by infinite series.
       

276. Gegenbauer biography
     
     • They are obtained from the hypergeometric series in certain cases where the series is in fact finite.
       

277. Levy Paul biography
     
     • He chose to attend the Ecole Polytechnique and he while still an undergraduate there published his first paper on semiconvergent series in 1905.
       
     • Not only did Levy contribute to probability and functional analysis but he also worked on partial differential equations and series.
       

278. Haret biography
     
     • Taking also into account commensurabilities, and using generalized Fourier series (which generate quasiperiodic solutions), Poincare proved the divergence of these series, which means instability, confirming in this way Haret's result.
       

279. Merrifield biography
     
     • He was editor of a series of scientific textbooks and he contributed to the series the volume Technical Arithmetic and Mensuration (1870).
       

280. Gillespie biography
     
     • He began to publish a series of important papers in the Cambridge Philosophical Society and in the Proceedings of the Edinburgh Mathematical Society.
       
     • In this RSE obituary Gillespie's books Integration (Edinburgh, 1939) and Partial Differentiation (Edinburgh, 1951), both published in Oliver & Boyd's series of university mathematical texts, are mentioned.
       

281. Cramer Harald biography
     
     • Also influenced by G H Hardy, Cramer's research resulted in the award of a PhD in 1917 for his thesis On a class of Dirichlet series.
       
     • He began to produce a series of papers on analytic number theory, and he addressed the Scandinavian Congress of Mathematicians in 1922 on Contributions to the analytic theory of numbers detailing his work on the topic up to that time.
       

282. Playfair biography
     
     • Playfair's simple and eloquent style consisted of a series of chapters clearly stating the Huttonian theory, giving the facts to support it, and the arguments given against it.
       
     • The second volume was entirely devoted to astronomy, while a third volume, which was intended to complete the series and cover the subjects of optics, electricity, and magnetism, was never completed.
       

283. Krieger biography
     
     • Her doctoral dissertation was On the summability of trigonometric series with localized properties - on Fourier constants and convergence factors of double Fourier series.
       

284. Paman biography
     
     • A "radical Quantity" is close to what today would be called a "variable", even though Paman implies that an expression can be composed of this variable only by taking powers of it, and by multiplicating by scalars, which means that Paman is thinking of polynomials or power series.
       
     • But we would not be able to define the derivative using Paman's terms since we consider more complicated functions than the polynomials or power series which Paman considered.
       

285. Krylov Aleksei biography
     
     • In a paper on forced vibrations of fixed-section pivots (1905), he presented an original development of Fourier's method for solving boundary value problems, pointing out its applicability to a series of important questions: for example, the theory of steam-driven machine indicators, the measurement of gas pressure in the conduit of an instrument, and the twisting vibrations of a roller with a flywheel on its end.
       
     • He studied the acceleration of convergence of Fourier series in a paper in 1912, and studied the approximate solutions to differential equations in a paper published in 1917.
       

286. Hankel biography
     
     • He also studied functions, now named Hankel functions or Bessel functions of the third kind, in a series of papers which appeared in Mathematische Annalen.
       
     • In the same way that he saw the importance of Grassmann's work, Hankel also must have considerable credit for seeing the importance of Bolzano's work on infinite series.
       

287. Boyle biography
     
     • The apparatus had been designed by Hooke and using it Boyle had discovered a whole series of important facts.
       
     • both a penetrating critique of Pascal's work on hydrostatics, full of acute observations upon Pascal's experimental method, and a presentation of a series of important and ingenious experiments on fluid pressure.
       

288. Gregory David biography
     
     • Gregory himself published Exercitatio geometria de dimensione curvarum in 1684 while at Edinburgh which was an interesting work developing his uncle's work on infinite series.
       
     • Gregory sent Newton a copy of his paper on infinite series, taking care to offer extensive praise to Newton.
       

289. Bolzano biography
     
     • Erste Lieferung (1810), the first of an intended series on the foundations of mathematics.
       
     • Bolzano wrote the second of his series but did not publish it.
       

290. Mitchell biography
     
     • A joint Mitchell/Fairweather paper, published in 1964, was the first in a series on high order alternating direction finite difference methods for elliptic PDE's.
       
     • Another conference was held in Dundee in 1969, and while in fact the first of what was to become a long series of biennial meetings associated with Dundee, the two earlier St Andrews meetings have quite properly been included and so have pushed it into third place.
       

291. Coulomb biography
     
     • He began to feel threatened by his political opponents in 1775 and began a series of reforms.
       
     • From examination of many physical parameters, he developed a series of two-term equations, the first term a constant and the second term varying with time, normal force, velocity, or other parameters.
       

292. Thompson John biography
     
     • The nonabelian finite simple groups fall into a small number of infinite series and 26 sporadic groups.
       
     • He gave a series on lectures on Galois groups at that meeting.
       

293. Chebyshev biography
     
     • Chebyshev submitted a paper on The calculation of roots of equations in which he solved the equation y = f (x) by using a series expansion for the inverse function of f.
       
     • This paper was on the convergence of Taylor series.
       

294. Delaunay biography
     
     • Delaunay found the longitude, latitude and parallax of the Moon as infinite series.
       
     • These gave results correct to 1 second of arc but were not too practical as the series converged slowly.
       

295. Gruenberg biography
     
     • Typical of this is his famous Some cohomological topics in group theory which appeared in the Queen Mary College Mathematics Notes series in 1967.
       
     • In 1976 he published Relation modules of finite groups which appeared in the Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics of the American Mathematical Society.
       

296. Askey biography
     
     • Twenty five years after he gave his series of ten lectures to the National Science Foundation Regional Conference, Askey published another major work on special functions.
       
     • He also gave two series of lectures in 1984, namely the University of Illinois Trjitzinsky Lectures and the Pennsylvania State University College of Science Lectures.
       

297. Milne William biography
     
     • However, Milne was also interested in mathematical education and published a series of papers and mathematical notes in the Mathematical Gazette.
       
     • These included: The geometrical meaning of the triad of points (1910); A property of the complete quadrangle (1911); The teaching of limits and convergence to scholarship candidates (1911); The teaching of limits and convergence to scholarship candidates (1912); The teaching of limits and convergence to scholarship candidates (1913); Another proof and generalisation of the theorem given in note 339 (1913); The teaching of modern analysis in secondary schools (1915); The graphical treatment of power series (1918); The uses and functions of a school mathematical library (1918); Mathematics and the pivotal industries (1919); The training of the mathematical teacher (1920); and Noether's canonical curves (1920).
       

298. Leibniz biography
     
     • On Huygens' advice, Leibniz read Saint-Vincent's work on summing series and made some discoveries of his own in this area.
       
     • While explaining his results on series to Pell, he was told that these were to be found in a book by Mouton.
       

299. Sokhotsky biography
     
     • His doctoral dissertation On definite integrals and functions with applications to expansion of series was an early investigation of the theory of singular integral equations.
       
     • His work is important in the development of the theory of functions, in particular having applications in the theory of hypergeometric series and differential equations.
       

300. Dixon Arthur biography
     
     • In 1908 Dixon began a series of publications on algebraic eliminants, carrying the subject forward from the point where Cayley had left it.
       
     • In the latter part of his career, Dixon published a series of around twelve joint papers with W L Ferrar on analytic number theory, summation formulae, Bessel functions and other topics in analysis.
       

301. Wintner biography
     
     • Hill's methods used infinite matrices and series expansions which he assumed were convergent but gave no proof.
       
     • These were Lectures on asymptotic distributions and infinite convolutions (1938), Analytical foundations of celestial mechanics (1941), Eratosthenian averages (1943), Theory of measure in arithmetical semigroups (1944), The Fourier transforms of probability distributions (1947), and An arithmetical approach to ordinary Fourier series (1945).
       

302. Jensen biography
     
     • Johan Ludwig Jensen's father was the sort of person who undertook a whole series of different projects yet, despite his good education and cultured style, the projects tended to end up financial failures.
       
     • He also studied infinite series, the gamma function and inequalities for convex functions.
       

303. Coulson biography
     
     • A mathematical account of the common types of wave motion in the Oliver and Boyd series.
       
     • He had maintained his links with the St Andrews applied mathematicians and he published another text in the Oliver and Boyd series, this one first appearing in 1948.
       

304. Schur biography
     
     • In a series of papers he introduced the concept now known as the 'Schur multiplier'.
       
     • Fifth, in divergent series; .
       

305. Harish-Chandra biography
     
     • Some major contributions by Harish-Chandra's work may be singled out: the explicit determination of the Plancherel measure for semisimple groups, the determination of the discrete series representations, his results on Eisenstein series and in the theory of automorphic forms, his "philosophy of cusp forms", as he called it, as a guiding principle to have a common view of certain phenomena in the representation theory of reductive groups in a rather broad sense, including not only the real Lie groups, but p-adic groups or groups over adele rings.
       

306. Luzin biography
     
     • I don't know how it happened, but I cannot be satisfied any more with analytic functions and Taylor series ..
       
     • He returned to Moscow in 1914 and he completed his thesis The integral and trigonometric series which he submitted in 1915.
       

307. Stueckelberg biography
     
     • In fact, these meals turned into a series of tutorials on the quantum mechanics.
       
     • Stueckelberg and I had laid out a whole series of calculations we wanted to attempt.
       

308. Walsh biography
     
     • However this may have been, Mr Walsh was for a series of years engaged in a constant endeavour to induce the principal learned societies of Europe to print his communications.
       
     • This merely contains a series of definitions and axioms, etc., beginning with the 'doctrine of ratio'.
       

309. Kolmogorov biography
     
     • However the person who made the deepest impression on Kolmogorov at this time was Stepanov who lectured to him on trigonometric series.
       
     • This was published jointly with Khinchin and contains the 'three series' theorem as well as results on inequalities of partial sums of random variables which would become the basis for martingale inequalities and the stochastic calculus.
       

310. Lions Jacques-Louis biography
     
     • In Paris he began a weekly numerical analysis seminar series and, later, he set up a numerical analysis laboratory.
       

311. Markov biography
     
     • Markov's early work was mainly in number theory and analysis, algebraic continued fractions, limits of integrals, approximation theory and the convergence of series.
       

312. Blaschke biography
     
     • He went to Italy, where he gave a series of lectures, before going on to Greece where again gave several lectures before returning to Germany on 16 April.
       

313. Burnside biography
     
     • This paper was the first of a series which Burnside described himself as follows (see for example [Pioneers of representation theory : Frobenius, Burnside, Schur, and Brauer (Providence, RI, 1999).',3)">3]):- .
       

314. Jonquieres biography
     
     • his results form a series of detailed supplements to the work of others and reflect Jonquieres's inventiveness in calculating rather than a more profound contribution to the advancement of the field.
       

315. Sturm biography
     
     • Papers of 1836-1837 by Sturm and Liouville on differential equations involved expansions of functions in series and is today well-known as the Sturm-Liouville problem, an eigenvalue problem in second order differential equations.
       

316. Thue biography
     
     • He wrote a many articles in series between 1906 and 1912 and he wrote in one of them:- .
       

317. Kempe biography
     
     • Kempe worked on the topic and presented a series of lectures at the Royal Institution on How to draw a straight line: A lecture on linkages in 1877.
       

318. Taylor James biography
     
     • When Mr Taylor took pen in hand, he showed that he possessed the gift of lucid expression, as is illustrated in the series of articles on "The Transit of Venus" in 1882, which he contributed to The Dollar Magazine of that year.
       

319. Wallis biography
     
     • About the beginning of my mathematical studies, as soon as the works of our celebrated countryman, Dr Wallis, fell into my hands, by considering the Series, by the Intercalation of which, he exhibits the Area of the Circle and the Hyperbola..
       

320. Gauss biography
     
     • His publications during this time include Disquisitiones generales circa seriem infinitam, a rigorous treatment of series and an introduction of the hypergeometric function, Methodus nova integralium valores per approximationem inveniendi, a practical essay on approximate integration, Bestimmung der Genauigkeit der Beobachtungen, a discussion of statistical estimators, and Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata.
       

321. Halmos biography
     
     • Halmos is known for both his outstanding contributions to operator theory, ergodic theory, functional analysis, in particular Hilbert spaces, and for his series of exceptionally well written textbooks.
       

322. Praeger biography
     
     • A series of three papers on a similar topic On the Sylow subgroups of a doubly transitive permutation group appeared in 1974 and 1975.
       

323. Pfaff biography
     
     • Pfaff did important work in analysis working on partial differential equations, special functions and the theory of series.
       

324. Lax Peter biography
     
     • SIAM published Lax's Hyperbolic systems of conservation laws and the mathematical theory of shock waves in their Conference Series in Applied Mathematics in 1973.
       

325. Lowenheim biography
     
     • Despite war service in France, Hungary and Serbia between August 1915 and December 1916, he published a series of important papers on mathematical logic during the eleven years from 1908 to 1919, extending work by Charles Peirce, Schroder, and Whitehead.
       

326. Abel biography
     
     • If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum had been rigorously determined.
       

327. Ostrogradski biography
     
     • His important work on ordinary differential equations considered methods of solution of non-linear equations which involved power series expansions in a parameter alpha.
       

328. Lexell biography
     
     • Lexell did work in analysis on topics other than differential equations, for example he suggested a classification of elliptic integrals and he worked on the Lagrange series.
       

329. Picard Emile biography
     
     • Picard's solution was represented in the form of a convergent series.
       

330. Copson biography
     
     • Copson was honoured by election to the Royal Society of Edinburgh in 1924 and was awarded the Keith Prize of the Society in 1941 for an outstanding series of papers published in the Proceedings.
       

331. Hensel biography
     
     • In 1897 the Weierstrass method of power-series development for algebraic functions led him to the invention of the p-adic numbers.
       

332. Mazurkiewicz biography
     
     • Although he only managed to recreate part of the work it was completed eleven years after his death and published as number 32 in the Mathematical Monographs series.
       

333. Beurling biography
     
     • Beurling worked on the theory of generalized functions, differential equations, harmonic analysis, Dirichlet series and potential theory.
       

334. Hopper biography
     
     • Compute the coefficients of the arctan series by next Thursday.
       

335. Moore Robert biography
     
     • This volume, published in the Colloquium Lectures Series of the American Mathematical Society, arose from the colloquium lectures which Moore gave in 1929 and is a self-contained introduction to the topic concentrating on Moore's own contributions to the subject.
       

336. Aryabhata I biography
     
     • It also contains continued fractions, quadratic equations, sums of power series and a table of sines.
       

337. Shnirelman biography
     
     • L A Lyusternik became a friend and important collaborator with Shnirelman and together they made significant contributions to topological methods in the calculus of variations in a series of paper written jointly between 1927 and 1929.
       

338. Whiston biography
     
     • While he was being subjected to charges of heresy he was bold enough to set out his religious beliefs in a series of pamphlets Primitive Christianity Revived (1711-12).
       

339. Pauli biography
     
     • This was in 1922, when he gave a series of guest lectures at Gottingen when he reported on his theoretical investigations on the periodic system of elements.
       

340. Olivier biography
     
     • In 1857, four years after Olivier died, Harvard University purchased 24 of Olivier's models from Fabre de Lagrange and after the university received the order Benjamin Peirce gave a series of lectures on the mathematics which they illustrated.
       

341. Knapowski biography
     
     • This came about in September 1956 when Turan gave a series of lectures on a new analytic method in Lublin.
       

342. Bjerknes Vilhelm biography
     
     • The next step forward in the mathematical approach was due to Richardson in 1922 when he reduced the complicated equations produced by Bjerknes's Bergen School to long series of simple arithmetic operations.
       

343. Padoa biography
     
     • Beginning in 1898 he gave a series of lectures at the Universities of Brussels, Pavia, Berne, Padua, Cagliari and Genoa.
       

344. Lyapunov biography
     
     • He returned to the problem that Chebyshev had placed before him and, in an extensive series of papers which continued until his death, developed the theory of figures of equilibrium of rotating heavy liquids.
       

345. Pic biography
     
     • The fourteen sections are: semigroups; divisibility in commutative semigroups; groups; semigroups of fractions; equivalences induced by a subgroup; conjugate subsets; inner automorphisms; Sylow subgroups; extensions of groups; direct products of subgroups; normal series; nilpotent groups; solvable groups; free semigroups and free groups.
       

346. Gergonne biography
     
     • Gergonne's first contributions to duality appear in a series of papers beginning in 1810.
       

347. Lipschitz biography
     
     • He carried out many important and fruitful investigations in number theory, in the theory of Bessel functions and of Fourier series, in ordinary and partial differential equations, and in analytical mechanics and potential theory.
       

348. Korteweg biography
     
     • Van der Waals was working on the phase separation of binary mixtures, and Korteweg supplied the necessary mathematical input with his work on folds on surfaces in a series of papers between 1891 and 1903, such as Over plooipunten en bijbehorende plooien in de nabijheid der randlijnen van het E'-vlak van Van der Waals (Plait points and corresponding plaits in the neighbourhood of the sides of the E'-surface of Van der Waals) (1902).
       

349. Hobson biography
     
     • His research concentrated on convergence, in particular convergence of series of orthogonal functions.
       

350. Wang Yuan biography
     
     • Wang Yuan fell in love with analytic number theory and gave a series of lectures to the graduate seminar based on Ingham's book The distribution of prime numbers.
       

351. Rosanes biography
     
     • He also wrote a series of papers on linearly dependent point systems in a plane and in space.
       

352. Anderson biography
     
     • From 1907 to 1915 he was A A Chuprov's assistant and his dissertation was on variance-difference methods for analysing time series.
       

353. Haselgrove biography
     
     • A great many terms of the Euler-Maclaurin or Riemann-Siegel series were used to calculate each entry.
       

354. Thomason biography
     
     • During the six years he spent there he produced a series of outstanding papers solving, among others, problems arising from Grothendieck's work in his paper with Berthelot and Illusie Theorie des Intersections et Theoreme de Riemann-Roch (1971).
       

355. Al-Biruni biography
     
     • These include: theoretical and practical arithmetic, summation of series, combinatorial analysis, the rule of three, irrational numbers, ratio theory, algebraic definitions, method of solving algebraic equations, geometry, Archimedes' theorems, trisection of the angle and other problems which cannot be solved with ruler and compass alone, conic sections, stereometry, stereographic projection, trigonometry, the sine theorem in the plane, and solving spherical triangles.
       

356. Chrystal biography
     
     • In [The Student (New Series) 4 (7) (3 December, 1890), 98.',26)">26] he is described by the Edinburgh students of 1890:- .
       

357. Koopmans biography
     
     • Koopmans, who lived in Hamden, Connecticut, died at Yale-New Haven Hospital after suffering a series of cerebral strokes.
       

358. Boersma biography
     
     • He published Computation of Fresnel integrals (1960) which gave a table of coefficients for approximation of Fresnel integrals by finite power series.
       

359. Simon biography
     
     • In a series of papers over the past ten years, Simon has developed methods for analysing this structure.
       

360. Ostrowski biography
     
     • One consequence of this association was his monograph Solution of equations and systems of equations which was published in 1960 and was the result of a series of lectures he had given at the National Bureau of Standards.
       

361. Aleksandrov biography
     
     • He laid the foundations of homology theory in a series of fundamental papers between 1925 and 1929.
       

362. Shannon biography
     
     • Working with John Riordan, Shannon published a paper in 1942 on the number of two-terminal series-parallel networks.
       

363. Johnson Barry biography
     
     • His mathematical publications started in 1964 with a series of papers on topological algebras, measure algebras and Banach algebras.
       

364. Dinghas biography
     
     • Dinghas produced a series of papers on isoperimetric problems in spaces of constant curvature.
       

365. Hartley biography
     
     • The Hartleys owned a series of tandems, but Mary [his wife] tended to fold her arms and sing as they went up hills, leaving Brian to do all the work.
       

366. Gluskin biography
     
     • he published a series of brilliant results on semigroups of linear transformations.
       

367. Tacquet biography
     
     • Tacquet introduced several ways of thinking which proved important in giving a foundation for future progress, for example in noting that one could pass from a finite progression to an infinite series.
       

368. Saint-Vincent biography
     
     • There are many topics covered in the book including a study of circles, triangles, geometric series, ellipses, parabolas and hyperbolas.
       

369. Pascal biography
     
     • From about this time Pascal began a series of experiments on atmospheric pressure.
       

370. Budan de Boislaurent biography
     
     • In total Budan published ten mathematical works and as one further example of his contributions we note that he submitted a paper on the summation of series to the Academy of Sciences in 1802 which was refereed by Biot and Lacroix.
       

371. Narayana biography
     
     • He used formulae and rules for the relations between magic squares and arithmetic series.
       

372. Foster biography
     
     • is a continuation of a long series of articles by [Foster] and his students which investigates unique factorization in certain classes of abstract algebras.
       

373. Luchins biography
     
     • Despite having to interrupt her doctoral studies, Luchins began to publish a series of papers with her husband including: Towards Intrinsic Methods in Testing (1946), A Structural Approach to the Teaching of the Concept of Area in Intuitive Geometry (1947), The Satiation Theory of Figural After-Effects and Gestalt Principles of Perception (1953), and Variables and Functions (1954).
       

374. Prager biography
     
     • In November and December 1954 Prager gave a series of lectures at the Polytechnic Institute in Zurich.
       

375. Calderon biography
     
     • Zygmund posed Calderon a question and the puzzled Calderon replied that the answer was contained in Zygmund's own book Trigonometric Series.
       

376. More Henry biography
     
     • He was a committed experimental scientist and he undertook a series of hydrostatic and pneumatic experiments to disprove Boyle's theory.
       

377. Thomson biography
     
     • This paper Fourier's expansions of functions in trigonometrical series was written to defend Fourier's mathematics against criticism from the professor of mathematics at the university of Edinburgh.
       

378. Cech biography
     
     • He continued this interest after the war ended and used his experiences with the school teachers to organise a series of school mathematics textbooks.
       

379. Lemaitre biography
     
     • In 1933 Einstein and Lemaitre gave a series of lectures in California.
       

380. Mostowski biography
     
     • He was the editor of the Mathematical, Astronomical and Physical series of the Bulletin of the Polish Academy of Sciences, on the editorial board of several journals including Fundamenta Mathematicae, Dissertationes Mathematicae, the Journal of Symbolic Logic and Studia Logica.
       

381. Smithies biography
     
     • Smithies early work was on integral equations and in 1958 his text Integral equations was published by Cambridge University Press in their Cambridge Tracts in Mathematics and Mathematical Physics Series.
       

382. Helmholtz biography
     
     • In mathematical appendices he advocated the use of Fourier series.
       

383. Mendelsohn biography
     
     • He wrote papers on a wide variety of combinatorial problems, for example: Symbolic solution of card matching problems (1946), Applications of combinatorial formulae to generalizations of Wilson's theorem (1949), Representations of positive real numbers by infinite sequences of integers (1952), A problem in combinatorial analysis (1953), The asymptotic series for a certain class of permutation problems (1956), and Some elementary properties of ill conditioned matrices and linear equations (1956).
       

384. Maupertuis biography
     
     • This work, and other work by Maupertuis on heredity, proposed a series of conjectures which some see as an early version of the theory of evolution.
       

385. Kochina biography
     
     • In 1996 she was awarded the M V Keldysh Gold Medal for a series of studies into hydrodynamics and the theory of filtration.
       

386. Peirce Charles biography
     
     • Peirce lectured on Pragmatism at Harvard in March to May of 1903 and published a series of essays explaining his ideas in The Monist in 1905.
       

387. Wirtinger biography
     
     • At the age of 71 he wrote the first of a series ground-breaking papers on higher dimensional spaces.
       

388. Klein biography
     
     • This work led him to consider elliptic modular functions which he studied in a series of papers.
       

389. Woods biography
     
     • He now published a whole series of papers - the next two were: A new relaxation treatment of flow with axial symmetry (1951), and The numerical solution of two-dimensional fluid motion in the neighbourhood of stagnation points and sharp corners (1952).
       

390. Dionis biography
     
     • Over a period of almost 20 years from 1764 he wrote a series of memoirs on eclipses, occultations (when one astronomical body comes in front of another), calculating orbits, and other such topics, and these were brought together in a two volume work Traite analytique des mouvements apparents des corps celestes which he published, volume one in 1786 and volume two in 1789.
       

391. Fowler biography
     
     • This work continued in a series of papers through the 1920s leading to the Adams Prize of the University of Cambridge in 1923-24 and was published in 1929 as the seminal volume, Statistical Mechanics, which had a second edition, minus the astrophysical applications, published in 1936.
       

392. Doob biography
     
     • In fact he undertook the work of writing the book because he had become intellectually bored while undertaking war work in Washington and so was enthusiastic when, in 1945, Shewhart invited him to publish a volume in the Wiley series in statistics.
       

393. Schutzenberger biography
     
     • Later he published a series of results on variable-length codes all of them reported in our book with Jean Berstel (Theory of Codes, Academic Press, 1984).
       

394. Oresme biography
     
     • Oresme also worked on infinite series and argued for an infinite void beyond the Earth.
       

395. Beaugrand biography
     
     • There is little wonder that he was not in favour among the French mathematicians for he had attacked the work of Desargues and published a series of pamphlets attacking the work of Descartes.
       

396. Merriles biography
     
     • He was founder of the Gramophone Club, an institution which will likely be permanent and which enables a select band of kindred spirits to discover, study and exploit the musical treasures that are hidden away in a series of gramophone records.
       

397. Maschke biography
     
     • Among the papers he published while at Chicago are: On systems of six points lying in three ways in involution (1896), Note on the unilateral surface of Mobius (1900), A new method of determining the differential parameters and invariants of quadratic differential quantics (1900), On superosculating quadric surfaces (1902), A symbolic treatment of the theory of invariants of quadratic differential quantics of n variables (1903), Differential parameters of the first order (1906); The Kronecker-Gaussian curvature of hyperspace (1906) and A geometrical problem connected with the continuation of a power-series (1906).
       

398. Weinstein biography
     
     • For example he solved Helmholtz's problem for jets, giving the first uniqueness and existence theorems for free jets in a series of papers from 1923 to 1929.
       

399. Bendixson biography
     
     • In this area he first studied uniform convergence of series of real functions and took an important step towards giving precise conditions when the limit function of continuous functions is continuous.
       

400. Bateman biography
     
     • He accumulated a vast store of information on all the familiar special functions and on his death the publication of his manuscripts was undertaken by Erdelyi and his associates in the form of the well-known series Higher Transcendental Functions and Tables of Integral Transforms.
       

401. Penrose biography
     
     • Beginning in 1959, Penrose published a series of important papers on cosmology.
       

402. Ringrose biography
     
     • He has written on operators of Volterra-type, compact linear operators, the Neumann series of integral operators, algebras of operators, automorphisms and derivations of operator algebras, and the cohomology of operator algebras.
       

403. Cremona biography
     
     • Also while at Bologna Cremona developed the theory of birational transformations, later known as Cremona transformations, and wrote a series of papers on twisted cubic surfaces.
       

404. Libri biography
     
     • These two sales of books imported from France contain a magnificent series of manuscripts and books by Galileo, Copernicus, Kepler, Cardan, etc., many with long notes pointing out their significance, and we must not allow ourselves to be blinded to the showmanship and originality of Libri's catalogue by his unenviable reputation as a forger and a thief.
       

405. Suzuki Michio biography
     
     • During this period he published a series of excellent papers: The lattice of subgroups of a finite group (in Japanese) (1950); On the finite group with a complete partition (1950); On the lattice of subgroups of finite groups (1951); On the L-homomorphisms of finite groups (1951); and A characterization of simple groups LF(2,p) (1951).
       

406. Aiken biography
     
     • He continued to work at Harvard on this series of machines, working next on the Mark III and finally the Mark IV up to 1952.
       

407. Andrews biography
     
     • The combinatorial and formal power series aspects of the subject have usually been treated in books on elementary number theory or combinatorial analysis.
       

408. Fuchs biography
     
     • In a series of papers (1880-81) Fuchs studied functions obtained by inverting the integrals of solutions to a second-order linear differential equation in a manner generalising Jacobi's inversion problem.
       

409. Frank biography
     
     • In mathematics he worked on the calculus of variations, Fourier series, function spaces, Hamiltonian geometrical optics, Schrodinger wave mechanics, and relativity.
       

410. Molyneux Samuel biography
     
     • Locke had written Some thought concerning education (1693) which was based on a series of letters he had written to Edward Clarke from Holland (where he had been in exile) advising him on how to bring up his son.
       

411. Rennie biography
     
     • In On dominated convergence he proves a converse of Lebesgue's theorem of dominated convergence and gives an application to Fourier series.
       

412. Hsu biography
     
     • During this period [at University College, London] Hsu wrote a remarkable series of papers on statistical inference which show the strong influence of the Neyman-Pearson point of view.
       

413. Werner biography
     
     • Perhaps of more interest is a series of twelve supplementary notes to this work.
       

414. Hammersley biography
     
     • He had already begun publishing statistical papers with The "effective" number of independent observations in an autocorrelated time series, a joint publication with G V Bayley apprearing in the Journal of the Royal Statistical Society in 1946.
       

415. Hoehnke biography
     
     • In this series of papers he investigates the structural relations between Brandt groupoids, Ehresmann groupoids, semigroups, Brandt semigroups, categories and groups.
       

416. Day biography
     
     • He even managed to travel to Connecticut to deliver a series of lectures.
       

417. Iwasawa biography
     
     • Iwasawa himself produced a series of deep papers throughout the 1960s which pushed his ideas much further.
       

418. Levi-Civita biography
     
     • Levi-Civita's work was of extreme importance in the theory of relativity, and he produced a series of papers elegantly treating the problem of a static gravitational field.
       

419. Dougall biography
     
     • This paper contains a new derivation of the coefficients in the expansion into a series of Legendre polynomials of the product of two Legendre polynomials.
       

420. Bochner biography
     
     • Bochner found that the Riemann Localisation Theorem was not valid for Fourier series of several variables (1935 - 1936), which led him indirectly to consider functions of several complex variables (1937).
       

421. La Hire biography
     
     • Bosse had published a series of works developing the geometric ideas that he had learnt from Desargues and had established his own school of art in 1661.
       

422. McClintock biography
     
     • A series of notebooks reflecting specific research into the Baskerville, Kemble, McClintock, and Wakeman families record the lines of descent through each ancestral surname.
       

423. Menabrea biography
     
     • In August 1840 Charles Babbage gave a series of lectures on his Analytical Engine at the Academy of Sciences in Turin.
       

424. Lions biography
     
     • Another major contribution by Lions, in a long series of important papers, is to variational problems.
       
     • He was even able to teach this material in the classroom, which was quite a pedagogical challenge! In a series of works begun with Evariste Sanchez-Palencia in 1995, he also developed the theory of 'sensitive problems', particularly as they arise in the theory of elastic shells.
       
     • In a long series of notes published in the Comptes Rendus until 2001, Lions returned to numerical analysis, and in particular to parallel computation and domain decomposition methods.
       

425. Walsh Joseph biography
     
     • He continued to publish a steady stream of papers with On the location of the roots of the derivative of a polynomial appearing in 1920 and then two papers A generalization of the Fourier cosine series and A theorem on cross-ratios in the geometry of inversion in 1921.
       

426. Stevin biography
     
     • With Prince Maurits now head of the army of the republic, and with Stevin as an advisor in his service, a series of military triumphs over the Spanish forces followed.
       

427. Stokes biography
     
     • I too feel that I have been thinking too much of late, but in a different way, my head running on divergent series, the discontinuity of arbitrary constants, ..
       

428. Vinogradov biography
     
     • However it was Vinogradov who, in a series of papers in the 1930s, brought the method to its full potential.
       

429. Zelmanov biography
     
     • At the Groups-St Andrews conference at Galway, Ireland in 1993, of which I [EFR] was a joint organiser, Zelmanov was one of the main speakers and he gave a series of five lectures on Nil rings methods in the theory of nilpotent groups.
       

430. Simson biography
     
     • 42 (1) (1991), 1-14.',3)">3], in which he discusses an early manuscript of Simson dealing with inverse tangent series and their use in calculating π.
       

431. Alfven biography
     
     • One result of these interests was a series of books that Hannes wrote, some together with Kerstin.
       

432. Al-Khwarizmi biography
     
     • This would not be worth mentioning if a series of conclusions about al-Khwarizmi's personality, occasionally even the origins of his knowledge, had not been drawn.
       

433. Poincare biography
     
     • He also showed that series expansions previously used in studying the 3-body problem were convergent, but not in general uniformly convergent, so putting in doubt the stability proofs of Lagrange and Laplace.
       

434. Cowling biography
     
     • During his time at Leeds the heavy load on a conscientious university professor with both departmental responsibilities and a commitment to scholarship took its toll; a series of health problems - a duodenal ulcer operation in 1954, a slipped disk in 1957 and a mild heart attack in 1960 - caused a slowing down of his activities well before his retirement.
       

435. Lovelace biography
     
     • She considered writing a long review, perhaps in the style of her Notes, of Ohm's work On galvanic series, mathematically determined but Babbage, who she looked to for encouragement, was becoming depressed at his own lack of success with financing the development of his computers and failed to give her the necessary support.
       

436. Young Alfred biography
     
     • He wrote a series of papers On quantitative substitutional analysis which arose out of the classical theory of invariants and contained his results in this area.
       

437. Wiener Norbert biography
     
     • Moreover, it led me very directly to the periodogram, and to the study of forms of harmonic analysis more general than the classical Fourier series and Fourier integral.
       

438. Thomson James biography
     
     • They inspired a series of rebellions in 1798 which were brutally repressed by the British.
       

439. Sampson biography
     
     • Sampson used a series of accurate observations from Harvard College Observatory to amend the existing theory of the satellite orbits, but the disagreement between theory and observation persisted.
       

440. Hausdorff biography
     
     • He introduced the concept of a partially ordered set and from 1901 to 1909 he proved a series of results on ordered sets.
       

441. Lanczos biography
     
     • Lanczos was much influenced by Fejer; he learnt from him about Fourier series, orthogonal polynomials, and interpolation.
       

442. Navier biography
     
     • He worked on applied mathematics topics such as engineering, elasticity and fluid mechanics and, in addition, he made contributions to Fourier series and their application to physical problems.
       

443. Mathieu Claude biography
     
     • In the same year, together with Biot, he embarked on a series of measurements of the length of the seconds pendulum at different points on the meridian, in particular at Bordeaux and at Dunkirk.
       

444. Clarke Joan biography
     
     • Joan Murray's greatest achievement was to establish the sequence of gold unicorns and heavy groats of James III and James IV, an extremely complex series which caused great difficulty for previous students.
       

445. Bondi biography
     
     • Also with co-authors, he wrote a series of papers Gravitational waves in general relativity.
       

446. Al-Banna biography
     
     • Other interesting results on summing series are the results .
       

447. Trudinger biography
     
     • This 1998 edition was reprinted in the "Classics in Mathematics" series by Springer-Verlag in 2001.
       

448. Lesniewski biography
     
     • From then until 1939 he published a series of twelve papers giving his theories of logic and mathematics.
       

449. Borok biography
     
     • Starting in the late 1960s, Valentina began a series of papers that lay the foundations for the theory of local and non-local boundary value problems in infinite layers for systems of partial differential equations.
       

450. Shtokalo biography
     
     • After 1945 he became particularly interested in the qualitative and stability theory of solutions of systems of linear ordinary differential equations in the Lyapunov sense and in the 1940s and 1950 published a series of articles and three monographs in these areas.
       

451. Cohen Wim biography
     
     • He worked on the problem of analysing congestion in telephone systems and from 1956 began publishing a series of papers.
       

452. Spanier biography
     
     • Spanier began joint work with Henry Whitehead and in a series of papers they introduced the method of duality in homotopy theory.
       

453. Dantzig biography
     
     • His most important work was in topological algebra and in addition to his doctoral thesis which we mentioned above, he wrote a whole series of papers on topological algebra.
       

454. Kovalevskaya biography
     
     • The paper on the reduction of abelian integrals to simpler elliptic integrals is of less importance but it consisted of a skilled series of manipulations which showed her complete command of Weierstrass's theory.
       

455. Merrilees biography
     
     • He was founder of the Gramophone Club, an institution which will likely be permanent and which enables a select band of kindred spirits to discover, study and exploit the musical treasures that are hidden away in a series of gramophone records.
       

456. Karsten biography
     
     • He also introduced Karsten to Euler and the two exchanged a series of letters (38 in all, of which 23 were written by Karsten) between 1758 and 1765.
       

457. Dyson biography
     
     • It gives us the opportunity to follow his own research about congruence properties of partitions, special series and infinite products, generating functions, and modular functions, ..
       

458. Hutton James biography
     
     • His simple and eloquent style consisted of a series of chapters clearly stating the Huttonian theory, giving the facts to support it, and the arguments given against it.
       

459. Chowla biography
     
     • He wrote on additive number theory (lattice points, partitions, Waring's problem), analysis, Bernoulli numbers, class invariants, definite integrals, elliptic integrals, infinite series, the Weierstrass approximation theorem), analytic number theory (Dirichlet L-functions, primes, Riemann and Epstein zeta functions), binary quadratic forms and class numbers, combinatorial problems (block designs, difference sets, Latin squares), Diophantine equations and Diophantine approximation, elementary number theory (arithmetic functions, continued fractions, and Ramanujan's tau function), and exponential and character sums (Gauss sums, Kloosterman sums, trigonometric sums).
       

460. Malcev biography
     
     • We mentioned some of the prizes he received above, such as the State Prize in 1946, but another important honour which he received in 1964 was a Lenin Prize for his series of papers on the applications of mathematical logic to algebra.
       

461. Fox Leslie biography
     
     • He contributed in many ways to promoting numerical analysis, for example in running summer schools, in developing links to industry, forming links with schools through the Mathematical Association, and with the writing of a wonderful series of books on the subject.
       

462. Adams Frank biography
     
     • He continued to produce work of outstanding depth and originality, and during his first few years at Manchester he wrote a series of papers On the groups J(X) which were highly influential in homotopy theory.
       

463. Berwald biography
     
     • Berwald wrote a series of major papers On Finsler and Cartan geometries.
       

464. Hua biography
     
     • In addition, Chinese Television (CCTV) produced a mini-series telling the story of Hua's life, which has been shown at least twice since then.
       

465. Lukacs biography
     
     • Jointly with Z W Birnbaum, he was the founding editor of the Academic Press Series in Probability and Mathematical Statistics (1962-85).
       

466. Sharp biography
     
     • In fact Sharp used Gregory's series with x = √3 to calculate π to 72 places, a task he had carried out in 1699.
       

467. Poncelet biography
     
     • The lectures he gave at Metz were first produced in lithographed form then, after a series of versions, were eventually published.
       

468. Edge biography
     
     • These have interesting geometrical properties and Edge investigated them in a series of papers spanning 40 years.
       

469. Cauchy biography
     
     • Cauchy was the first to make a rigorous study of the conditions for convergence of infinite series in addition to his rigorous definition of an integral.
       

470. Lasker biography
     
     • While in England he gave a series of lectures on chess which he wrote up for publication as Common Sense in Chess.
       

471. Schmidt biography
     
     • He also expanded functions related to the integral of the kernel function as an infinite series in a set of orthonormal eigenfunctions.
       

472. Suetuna biography
     
     • This book, based mainly on the Riemann zeta-functions and L-functions, is a unique exposition of the analytical theory of numbers in a modern sense as can be seen from the chapter headings: I) Riemann's zeta-functions; II) Hecke's L-functions; III) Dirichlet's L-functions; and IV) Artin's L-series.
       

473. Rado Richard biography
     
     • Some of his more minor work was in topics such as the convergence of sequences and series.
       

474. Gupta biography
     
     • The first of these is Multiple decision procedures: theory and methodology of selecting and ranking populations written jointly with S Panchapakesan and published in the Wiley Series in Probability and Mathematical Statistics in 1979.
       

475. Whitehead Henry biography
     
     • The long series of collaborative papers written between 1950 and 1960 reflects his eagerness to share his ideas and to interest himself in the results of others, which remained undiminished to the end of his life.
       

476. Moore Eliakim biography
     
     • He also studied infinite series of finite simple groups.
       

477. Brauer Alfred biography
     
     • From the late 1950s Brauer published a series of papers on nonnegative matrices, a topic studied by Frobenius towards the end of his career.
       

478. Lewy biography
     
     • he published a series of fundamental papers on partial differential equations and the calculus of variations.
       

479. Pontryagin biography
     
     • He then produced a series of papers on differential games which extends his work on control theory.
       

480. Delambre biography
     
     • presents each major chronological period in a series of discrete analyses of one treatise after another.
       

481. Bartel biography
     
     • He lectured on this topic and his lecture series was published.
       

482. Monge biography
     
     • Over the next few years he submitted a series of important papers to the Academie on partial differential equations which he studied from a geometrical point of view.
       

483. Wiener Christian biography
     
     • For example, he used imaginary projection and developed a grid method that can be derived from the theory of cyclically projected point series.
       

484. Amsler biography
     
     • adapted easily to the determination of static and inertial moments and to the coefficients of Fourier series: it proved especially useful to shipbuilders and railway engineers.
       

485. Bruno Giordano biography
     
     • Oxford seemed a place of learning that looked attractive to Bruno who visited there in the summer of 1583 and gave a series of lecturers on Copernicus's theory that the Earth rotated round the fixed Sun.
       

486. Bernoulli Daniel biography
     
     • Thus, in one stroke he derived the entire series of such curves as the velaria, lintearia, catenaria..
       

487. John biography
     
     • He wrote an important series of papers on numerical analysis, studying ill-posed problems.
       

488. Sierpinski biography
     
     • His work on functions of a real variable include results on functional series, differentiability of functions and Baire's classification.
       

489. De Vries Henrik biography
     
     • These culminated in a series of articles in the Nieuw Tijdschrift voor Wiskunde (New Journal of Mathematics), which were later collected, together with some other items, in a three volume publication entitled 'Historische Studien' (1926).
       

490. Brahmagupta biography
     
     • Rules for summing series are also given.
       

491. Laszlo biography
     
     • We must not give the impression that Laszlo's only research interest was in the history of mathematics for he also published a long series of papers on fuzzy groups, some written with his collaborator Iulius Gyula Maurer, beginning in 1987.
       

492. Angeli biography
     
     • James Gregory studied with Angeli in Padua from 1664 to 1668 and learnt from him about series expansions of functions.
       

493. Ingham biography
     
     • Ingham's work was on the Riemann zeta function, the theory of numbers, the theory of series and Tauberian theorems.
       

494. Slutsky biography
     
     • He also studied correlations of related series for a limited number of trials.
       

495. Moser William biography
     
     • The full collection, corrected and improved, has been published by Mathematics Magazine in its spectrum series as "500 Mathematical Challenges" in 1995.
       

496. Lehmer Derrick N biography
     
     • Under each tooth in this second series of gears is a small hole.
       

497. Hermite biography
     
     • The letters he exchanged with Jacobi show that Hermite had discovered some differential equations satisfied by theta-functions and he was using Fourier series to study them.
       

498. Ricci-Curbastro biography
     
     • The first was a series of articles on Maxwell's theory of electrodynamics and the work of Clausius which Betti asked him to write.
       

499. Thompson Robert biography
     
     • As a result he published a number of series of papers attacking particular problems.
       

500. Warschawski biography
     
     • In 1955 he published two papers in Experiments in the computation of conformal maps published in the National Bureau of Standards Applied Mathematics Series.
       

501. Goodstein biography
     
     • Educated at St Paul's School London, Louis won scholarships and a prize for an essay on divergent series.
       

502. Hellinger biography
     
     • Hellinger's position at Evanston throughout the war was precarious with a series of one-year appointments but he acquired American citizenship in 1944 and worked at Evanston until 1949 when he retired.
       

503. Alcuin biography
     
     • These were a series of illuminated masterpieces written largely in gold, often on purple coloured vellum.
       

504. Grauert biography
     
     • This text is an excellent introduction to the classical themes of modern several complex variables theory: domains of holomorphy, holomorphic complexity, pseudoconvexity, the ring of convergent power series, analytic subvarieties and the several variables version of the Mittag-Leffler and Weierstrass problems ..
       

505. Kahler biography
     
     • At first he was inspired by books his mother bought him about Sven Hedin, a Swedish explorer who led a series of expeditions through Central Asia and as a consequence made important archaeological and geographical findings.
       

506. Pappus biography
     
     • It seems likely that this work was not originally written as a single treatise but rather was written as a series of books dealing with different topics.
       

507. Borda biography
     
     • He also developed a series of trigonometric tables in conjunction with his surveying techniques.
       

508. Veblen biography
     
     • He was the Colloquium Lecturer for the Society in 1916 when he gave a series of lectures on topology.
       

509. Bernoulli Nicolaus(I) biography
     
     • In his letters to Euler (1742-43) he criticises Euler's indiscriminate use of divergent series.
       

510. Laurent Pierre biography
     
     • Cauchy reported on Laurent's entry Memoire sur le calcul des variations, which contains the Laurent series for a complex function, on 20 May 1843.
       

511. David biography
     
     • The first investigates the behaviour of the nonparametric test for randomness based on the number of alternations of an event and its complement in a series of trials.
       

512. Koszul biography
     
     • In the autumn of 1958 he again held a seminar series in São Paulo, this time on symmetric spaces.
       

513. Frohlich biography
     
     • In the present paper the fields of at most class two over the rational field are studied, the class of a field being defined as the length of the central series of the Galois group.
       

514. Carleman biography
     
     • Carleman is now remembered for remarkable results in integral equations (1923), quasi-analytic functions (1926), harmonic analysis (1944), trigonometric series (1918-23), approximation of functions (1922-27) and Boltzmann's equation (1944).
       

515. Liouville biography
     
     • Sturm and Liouville examined general linear second order differential equations and examined properties of their eigenvalues, the behaviour of the eigenfunctions and the series expansion of arbitrary functions in terms of these eigenfunctions.
       

516. Sundman biography
     
     • The most famous contribution of Sundman was his solution of the three-body problem which he accomplished using analytic methods to prove the existence of an infinite series solution.
       

517. Feynman biography
     
     • Returning to their respective homes in the summer of 1936 the two exchanged a series of remarkable letters as they tried to develop a version of space-time where (quoted from one of the letters - see [Genius : The Life and Science of Richard Feynman (New York, 1992).',6)">6]):- .
       

518. Lupas biography
     
     • For example On Bernstein power series (1966) was reviewed by D E Wulbert who wrote:- .
       

519. Enskog biography
     
     • Enskog used Hilbert's methods to work out a series expansion of the velocity distribution function and wrote this up for his doctoral dissertation at Uppsala in 1917.
       

520. Brahe biography
     
     • In fact, however, their construction can be traced in his logs and rationalized as several series of experiments which only produced his major instruments in the mid-1580's.
       

521. Frege biography
     
     • a series of brilliant philosophical articles in which he elaborated his philosophy of logic.
       

522. Poisson biography
     
     • His approach to these problems was to use series expansions to derive approximate solutions.
       

523. Cherry Colin biography
     
     • He was honoured in 1987 when Imperial College inaugurated 'The Colin Cherry Memorial Lecture.' The information given about the Memorial Lecture series gives the following description of Cherry's contributions:- .
       

524. Wielandt biography
     
     • Where Hall had started from arithmetical questions and product decompositions, my own work was triggered by a question of Robert Remak of a quite different type: is the group generated by two subgroups that occur in composition series always of the same kind? In my Habilitationsschrift I expanded the discovery that this question can be answered affirmatively to a detailed study of the normal structure of finite groups.
       

525. Jeffery Ralph biography
     
     • His presidential address to the Society was Trigonometric series which he gave in 1953 and three years later it was published as a 39 page book by the University of Toronto Press.
       

526. Stone biography
     
     • For example he published An unusual type of expansion problem (1924), A comparison of the series of Fourier and Birkhoff (1926), Developments in Legendre polynomials (1926), and Developments in Hermite polynomials (1927).
       

527. Privat de Molieres biography
     
     • He also published a series of Memoirs of the Academy and in several articles in the Journal de Trevoux.
       

528. Landau biography
     
     • He submitted this habilitation thesis in 1901, only two years after his doctorate, consisted of his work on Dirichlet series, a topic in analytic number theory.
       

529. Kruskal Martin biography
     
     • Analysing asymptotic series also led Kruskal to become interested in surreal numbers, generalisations of real numbers introduced by John Conway.
       

530. Sitter biography
     
     • He published a series of papers (1916-17) on the astronomical consequences of Einstein's general theory of relativity.
       

531. Gram biography
     
     • He published a paper on these topics On series expansions determined by the methods of least squares and for this work he was awarded the degree of Doctor of Science in 1879.
       

532. Hopkins biography
     
     • referred to a series of important experiments which he had instituted at Manchester with the advice of Sir William Thomson and the assistance of Messrs Joule and Fairbairn, to determine the temperature of melting substances under great pressure.
       

533. Tinseau biography
     
     • He published a series of anti-Revolution writings from 1792 onwards and tried to organise uprisings in France, as did Charles-Philippe who made an unsuccessful attempt to land in the Vendee to lead a royalist rising there.
       

534. Deligne biography
     
     • The areas on which he has worked, in addition to algebraic geometry, are Hilbert's 21st problem, Hodge theory, theory of moduli, modular forms, Galois representations, L-series and the Langlands conjectures, and representations of algebraic groups.
       

535. Sridhara biography
     
     • There are sections of the book devoted to arithmetic and geometric progressions, including progressions with a fractional numbers of terms, and formulae for the sum of certain finite series are given.
       

536. Lorentz biography
     
     • In 1909 he published his "Theory of Electrons", based on a series of lectures at Columbia University, and in 1916 he published in French at Leipzig an account of statistical thermodynamic theories, based on lectures delivered at the College de France in 1912.
       

537. Godel biography
     
     • In 1934 Godel gave a series of lectures at Princeton entitled On undecidable propositions of formal mathematical systems.
       

538. Collingwood biography
     
     • he had great intellectual powers which enabled him to achieve excellence in diverse activities conducted in parallel and not in series.
       

539. Lissajous biography
     
     • At the conclusion of this beautiful series of experiments, which, thanks to the skill of those who performed them, were all successful, on the motion of Mr Faraday, the thanks of the meeting were unanimously voted to M M Lissajous and Duboscq and communicated to those gentlemen by his Grace the President, The Duke of Northumberland.
       

540. De Vries Hendrik biography
     
     • These culminated in a series of articles in the Nieuw Tijdschrift voor Wiskunde (New Journal of Mathematics), which were later collected, together with some other items, in a three volume publication entitled 'Historische Studien' (1926).
       

541. Whewell biography
     
     • Whewell entered Cambridge in October 1812, but by this time the family had suffered a series of tragedies with his mother dying in 1807 and three of his younger brothers dying before William began his university studies.
       

542. Lagny biography
     
     • De Lagny is well known for his contributions to computational mathematics, calculating π to 120 places and also making useful comments on the convergence of the series he was using.
       

543. Grassmann biography
     
     • After writing a series of articles on constitutional law, Grassmann became increasingly at odds with the political direction the newspaper was going and withdrew form it.
       

544. Dunbar biography
     
     • She published the paper Absorption and scattering of X-rays and the characteristic radiations of the J series in collaboration with Charles Glover Barkla.
       

545. Kato biography
     
     • The Department of Mathematics of the University of California, Berkeley, published Kato's notes Quadratic forms in Hilbert spaces and asymptotic perturbation series in 1955.
       

546. Puissant biography
     
     • The map was produced with considerable detail, the projection used spherical trigonometry, truncated power series and differential geometry.
       

547. Robinson Raphael biography
     
     • In a series of papers Robinson showed that a number of mathematical theories are undecidable.
       

548. Kirkman biography
     
     • This was followed by a series of papers.
       

549. Kepler biography
     
     • From the first, Kepler had sought a rule relating the sizes of the orbits to the periods, but there was no slow series of steps towards this law as there had been towards the other two.
       

550. Fermi biography
     
     • In the summer of 1954 Fermi returned to Italy and gave a series of lectures in the Villa Monastero in Varenna on Lake Como.
       

551. Segre Beniamino biography
     
     • He gave a series of lectures in London in 1950 which were published as Arithmetical questions on algebraic varieties in 1951.
       

552. Severi biography
     
     • His doctoral thesis, together with a series of other papers which Severi published at this time, deal with enumerative geometry, a subject which had been started by Schubert.
       

553. Bowen biography
     
     • In 1975 Bowen published Equilibrium states and the ergodic theory of Anosov diffeomorphisms in the Springer Lecture Notes in Mathematics Series.
       

554. Roy biography
     
     • In 1970 the book Essays in probability and statistics, edited by R C Bose, I M Chakravarti, P C Mahalanobis, C R Rao and K J C Smith, dedicated to his memory was published in the University of North Carolina Monograph Series in Probability and Statistics.
       

555. Dickstein biography
     
     • In 1884 he was one of the two founders of a series of mathematics and physics textbooks which were written in Polish.
       

556. Al-Khujandi biography
     
     • During the year 994 al-Khujandi used the very large instrument to observe a series of meridian transits of the sun near the solstices.
       

557. Goldie Alfred biography
     
     • During his retirement he published papers such as (with Gunter Krause) Associated series and regular elements of Noetherian rings (1987), Rings with an additive rank function (1990), and (with Gunter Krause) Embedding rings with Krull dimension in Artinian rings (1996).
       

558. Ince biography
     
     • The nucleus of an integral equation for one of the periodic Lame functions is expanded in series of products of the characteristic functions ..
       

559. Landen biography
     
     • Landen wrote on dynamics, summation of series and an important transformation giving a relation between elliptic functions.
       

560. Rittenhouse biography
     
     • This work was not new in the sense that had he known of Taylor series published in 1717 he could have deduced his results easily.
       

561. Helly biography
     
     • He taught in a Gymnasium, gave private tuition, and wrote solution manuals for a series of standard textbooks.
       

562. Rado biography
     
     • He gave his series of talks on his major contributions on surface area.
       

563. Neumann Bernhard biography
     
     • In 1955 when I first arrived in Manchester to work with B H Neumann he suggested that I read his paper 'Ascending derived series' which had only just been submitted for publication.
       

564. Redei biography
     
     • The classical approach to the study of p-groups consists in the investigation of their subgroups and central series.
       

565. Lamy biography
     
     • Next came Demonstration de la verite et de la saintete de la morale chretienne (1688), and Harmonia sive Concordia quatuor evangelistarum, in qua vera series actuum et sermonum Domini nostri (1689).
       

566. Newman biography
     
     • A series of papers by Newman on this topic between 1926 and 1932 revolutionised the field.
       

567. Bruns biography
     
     • He worked on the three-body problem showing that the series solutions of the Lagrange equations can change between convergent to divergent for small perturbations of the constants on which the coefficients of the time depend.
       

568. Netto biography
     
     • There he taught courses on advanced algebra, the calculus of variations, mechanics, Fourier series, and synthetic geometry.
       

569. Cataldi biography
     
     • Cataldi found square roots of numbers by use of an infinite series leading to an early investigation into continued fractions.
       
     • In this work the square root of a number is found through the use of infinite series and unlimited continued fractions.
       

570. Morawetz biography
     
     • In a series of three significant papers in the late 1950s, Cathleen Morawetz used functional analysis coupled with ingenious new estimates for an equation of mixed type, i.e.
       

571. Wiles biography
     
     • He filled what he thought were the remaining few gaps and gave a series of lectures at the Isaac Newton Institute in Cambridge ending on 23 June 1993.
       

572. Urysohn biography
     
     • He published a series of short notes on this topic during 1922.
       

573. Fuss biography
     
     • Most of Fuss's papers are solutions to problems posed by Euler on spherical geometry, trigonometry, series, differential geometry and differential equations.
       

574. Rankin biography
     
     • With his usual fine sense of history, [Rankin] begins the discussion, not with Ramanujan himself, but rather with the older English mathematician J W L Glaisher (born in 1848), who initiated the study of multiplicative properties of the Fourier coefficients of modular forms in his series of papers, published in 1907, dealing with ..
       

575. Cohen biography
     
     • His doctoral thesis Topics in the Theory of Uniqueness of Trigonometric Series was written under the supervision of Antoni Zygmund.
       

576. Buffon biography
     
     • His main mathematical contribution of this period was the publication of his translation of Newton's Method of Fluxions and infinite series in 1740.
       

577. Littlewood Dudley biography
     
     • Another reason was certainly the work of Hilbert, but Littlewood tried to remedy the "tensor reason" in a series of papers on tensors and invariant theory.
       

578. Burali-Forti biography
     
     • In 1893-94 Burali-Forti gave an informal series of lectures on mathematical logic at the University of Turin.
       

579. Schwarz biography
     
     • In answering the problem of when Gauss's hypergeometric series was an algebraic function Schwarz, as he had done so many times, developed a method which would lead to much more general results.
       

580. Matsushima biography
     
     • His research in Osaka took a somewhat different direction and he wrote a series of papers on cohomology of locally symmetric spaces.
       

581. Tits biography
     
     • He followed this with three series of lectures on the following topics .
       

582. Paramesvara biography
     
     • Paramesvara made a series of eclipse observations between 1393 and 1432 which we have referred to above.
       

583. Horner biography
     
     • Horner made other mathematical contributions, however, publishing a series of papers on transforming and solving algebraic equations, and he also applied similar techniques to functional equations.
       

584. Hutton biography
     
     • In 1776 he published A new and general method of finding simple and quickly converging series and two year later, in the same Transactions he published The force of fired gunpowder and the velocity of cannon balls.
       

585. Ito biography
     
     • In 1960 Ito visited the Tata Institute in Bombay, India, where he gave a series of lectures surveying his own work and that of other on Markov processes, Levy processes, Brownian motion and linear diffusion.
       

586. Bouvard biography
     
     • He also had at his disposal two fine series of post-discovery observations, one by the Paris observatory, the other by the Greenwich observatory.
       

587. Enriques biography
     
     • He produced a series of papers over a period of 20 years which, together with Castelnuovo, finally produced a classification of algebraic surfaces.
       

588. Borel Armand biography
     
     • In the summer of 1951 he gave a series of lectures in Zurich on the Leray's ideas on the theory of homological invariants of locally compact spaces and of continuous mappings which was published as a 95 page book of mimeographed notes with the title Cohomologie des espaces localement compacts, d'apres J Leray.
       

589. Edgeworth biography
     
     • In 1892 Edgeworth examined correlation and methods of estimating correlation coefficients in a series of papers.
       

590. Metzler biography
     
     • Furthermore, he was the one who first pointed out that one could have the transcendental functions of a square matrix simply by substituting it into the appropriate Taylor series.
       

591. Zassenhaus biography
     
     • In a long series of papers he applied Lie algebras to problems of theoretical physics.
       

592. Montroll biography
     
     • During his time at Maryland, Montroll published Topics in statistical mechanics of interacting particles which was 86 pages of mimeographed notes of a lecture series, written jointly with G F Newell.
       

593. Vallee Poussin biography
     
     • Most of the additional material appeared in small type and covered topics such as set theory, in particular the Schroder-Bernstein theorem, the Lebesgue integral, functions of bounded variation, the Jordan curve theorem, polynomial approximation, Parseval's theorem on trigonometric series, results of Fejer, etc.
       

594. Quine biography
     
     • And corresponds to terminals in series, or to those in parallel, so that if you simplify mathematical logical steps, you have simplified your wiring.
       

595. Szekeres biography
     
     • In 1965 he wrote a numerical analysis paper Some estimates of the coefficients in the Chebyshev series expansion of a function and a paper dealing with a combinatorial problem On a problem of Schutte and Erdos written jointly with his wife Esther.
       

596. Apollonius biography
     
     • Included in it are a series of propositions which, though worked out by the purest geometrical methods, actually lead immediately to the determination of the evolute of each of the three conics; that is to say, the Cartesian equations of the evolutes can be easily deduced from the results obtained by Apollonius.
       

597. Agnesi biography
     
     • In 1738 she published Propositiones Philosophicae a series of essays on philosophy and natural science.
       

598. Turner John biography
     
     • At Easter Mr John Turner, Deputy Rector and Principal Teacher of Mathematics, retired after forty years' service, and the best memorial of his work is the unbroken series of success gained by his pupils in Mathematics, and their abiding gratitude.
       

599. Todhunter biography
     
     • Still less can I imagine how it came to pass that he published a whole series of excellent mathematical works.
       

600. Arnauld biography
     
     • Pascal wrote a series of 18 letters now known as Les Provinciales during the years 1656 and 1657 in defence of Arnauld.
       

601. Von Neumann biography
     
     • In the second half of the 1930's and the early 1940s von Neumann, working with his collaborator F J Murray, laid the foundations for the study of von Neumann algebras in a fundamental series of papers.
       

602. Mercer biography
     
     • Mercer was a mathematical analyst of originality and skill; he made noteworthy advances in the theory of integral equations, and especially in the theory of the expansion of arbitrary functions in series of orthogonal functions.
       

603. Chen biography
     
     • Kuo-Tsai Chen is best known to the mathematics community for his work on iterated integrals and power series connections in conjunction with his research on the cohomology of loop spaces.
       

604. Ramus biography
     
     • Using this approach Ramus worked on many topics and wrote a whole series of textbooks on logic and rhetoric, grammar, mathematics, astronomy, and optics.
       

605. Hamilton William biography
     
     • Hamilton was one of the first in a series of British logicians to create the algebra of logic and introduced the 'quantification of the predicate'.
       

606. Turan biography
     
     • As regards the latter, Turan found new approaches to such topics as quasi-analytic classes, Fabry's gap theorem and the theory of lacunary series, amongst others.
       

607. Deuring biography
     
     • On this second visit he gave a series of lectures which were published as Lectures on the theory of algebraic functions of one variable in 1973.
       

608. Zuse biography
     
     • In fact Zuse designed several computers other than those of his Z series.
       

609. Calugareanu biography
     
     • Returning to Weierstrass's point of view on analyticity and the definition of analytic functions via Taylor series elements, Calugareanu created the theory of invariants and covariants of analytic continuation.
       

610. Frechet biography
     
     • The book Lecons sur les fonctions de variables reelles et les developpements en series de polynomes was published in 1905.
       
     • This campaign took the unusual form of a survey sent to colleagues all over the world as well as a series of papers, committee reports, and censure motions within the International Institute of Statistics.
       

611. Magnus biography
     
     • During this period Magnus introduced Lie ring methods to study the lower central series of free groups.
       

612. Bernays biography
     
     • Between 1937 and 1954 Bernays wrote a whole series of articles in the Journal of Symbolic Logic which attempted to achieve this goal.
       

613. Van Lint biography
     
     • The Dutch had assumed that they would be able to return to the pre-war situation in Indonesia, but an Independence movement made this impossible and a series of risings marked the beginning of a revolution.
       

614. Dieudonne biography
     
     • He was one of the main contributors to the Bourbaki series of texts from the time that the group came into existence and in many ways he was the leading influence in a group whose whole object was to avoid anyone taking on this role.
       

615. Moser Jurgen biography
     
     • First we mention Lectures on Hamiltonian systems (1968) which examines problems of the stability of solutions, the convergence of power series expansions, and integrals for Hamiltonian systems near a critical point.
       

616. Bolyai Farkas biography
     
     • His study of the convergence of series includes a test equivalent to Raabe's test which he discovered independently and at about the same time as Raabe.
       

617. Taylor Geoffrey biography
     
     • At the age of 11 he attended a series of children's Christmas lectures on The principles of the electric telegraph and these made a strong impression on him.
       

618. Iyanaga biography
     
     • This came about through talking to the Departmental Assistant T Shimizu who discussed with him questions about power series.
       

619. Fields biography
     
     • The series of International Congresses of Mathematicians began in Zurich in 1897 but no congress was held during World War I (1914-18).
       

620. Whitehead biography
     
     • Science and the Modern World (1925), a series of lectures given in the United States, served as an introduction to his later metaphysics.
       

621. Rolle biography
     
     • It amplified the concepts of limits of roots of equations, provided the fundamentals from which Maclaurin derived his formula, began modern methods of series for determining roots, and discussed the relationship of imaginary roots in equations and their derivatives.
       

622. Hudde biography
     
     • In 1656 he gave the power series expansion of ln(1+x).
       

623. Straus biography
     
     • In 1949 Straus collaborated with Richard Bellman publishing Continued fractions, algebraic functions and the Pade table in which they gave a method for obtaining the rational approximants of Frobenius-Pade for power series expansions of algebraic functions.
       

624. Olive biography
     
     • She published papers such as Binomial functions and combinatorial mathematics (1979), A combinatorial approach to generalized powers (1980), Binomial functions with the Stirling property (1981), Some functions that count (1983), Taylor series revisited (1984), Catalan numbers revisited (1985), A special class of infinite matrices (1987), and The ballot problem revisited (1988).
       

625. Zeno of Elea biography
     
     • Although they have often been dismissed as logical nonsense, many attempts have also been made to dispose of them by means of mathematical theorems, such as the theory of convergent series or the theory of sets.
       

626. Chree biography
     
     • Series A, Containing Papers of a Mathematical and Physical Character Vol.
       

627. Atkinson biography
     
     • In fact much of his early research followed on from this beginning with papers such as A summation formula for p(n), the partition function (1939), The mean value of the zeta-function on the critical line (1941), A divisor problem (1941), The Abel summation of certain Dirichlet series (1948), A mean value property of the Riemann zeta-function (1948), The mean-values of arithmetical functions (1949), and The mean-value of the Riemann zeta function (1949).
       

628. Dickson biography
     
     • Dickson's search for a counterexample led him to consider non-associative algebras and in a series of papers he determined all three and four-dimensional (non-associative) division algebras over a field.
       

629. Rudin biography
     
     • In August 1974 Rudin gave a series of lectures on set theoretic topology at the CBMS Regional Conference held at the University of Wyoming, Laramie.
       

630. Shoda biography
     
     • develops generalized theories of normal chains, composition series, of direct and subdirect products, and generalizations of the Jordan-Holder and the Remak-Schmidt-Ore theorems.
       

631. Sonin biography
     
     • He obtained a Master's Degree with a thesis on the expansion of functions in infinite series submitted in 1871.
       

632. Franklin biography
     
     • He worked on the four colour problem and also published books on calculus, differential equations, complex variable and Fourier series.
       

633. Brauer biography
     
     • In 1949 Brauer was awarded the Cole Prize from the American Mathematical Society for his paper On Artin's L-series with general group characters which he published in the Annals of Mathematics in 1947.
       

634. Todd John biography
     
     • He became ill while giving a lecture series on group theory and quantum mechanics.
       

635. Wazewski biography
     
     • His interest in that topic began around 1960 and he published a series of important papers on the topic through the 1960s.
       

636. Semple biography
     
     • The result was a series of fascinating papers and one further book Generalized Clifford parallelism (1971).
       

637. Singer biography
     
     • Singer's series of five papers with Michael F Atiyah on the Index Theorem for elliptic operators (which appeared in 1968 - 71) and his three papers with Atiyah and V K Patodi on the Index Theorem for manifolds with boundary (which appeared in 1975 - 76) are among the great classics of global analysis.
       

638. Gleason biography
     
     • Chapters I to VI cover elementary logic and set theory; Chapters VII to X deal with the various "number systems" from the natural integers to the complex numbers; Chapter XI briefly returns to set theory (countable sets, cardinal numbers and the axiom of choice); finally, the last four chapters deal, respectively, with limits of complex sequences, infinite series and products, metric spaces, and the elementary theory of holomorphic functions of one variable (Cauchy integral excluded, but the logarithmic function is defined and studied).
       

639. Plateau biography
     
     • He then carried out a series of experiments repeating the original accident but also investigating the shape of the drops of oil when the mixture of water and alcohol is rotating.
       

640. Peterson biography
     
     • Peterson's most important paper was 'On the ratios and relationships between curved surfaces' (1866), devoted to deformation of surfaces, which laid the foundation for a series of papers on the problem of bending on a principal basis, i.e., preserving the conjugacy of a certain net on the surface, the first example of which for deformation of surfaces of revolution on a surface of revolution was found by Minding ..
       

History Topics

1. Bolzano publications.html
   
   • This is the first volume of the new series (which is still being produced) of Bernard Bolzano- Gesamtausgabe.
     
   • This volume in the series of posthumous writings of the Collected works of Bernard Bolzano contains transcriptions of some early manuscripts which present his views in the 1810's on the foundations of logic, mathematics and physics.
     
   • Contains his thoughts on Euclidean geometry, manipulations of series, functions and foundations of calculus, and topics in mechanics.
     
   • Band 11 Teil 1 (German), [Bernard Bolzano - collected works: Series I.
     
   • Band 11 Teil 3 (German), [Bernard Bolzano - collected works: Series I.
     
   • Band 12 Teil 1 (German), [Bernard Bolzano - collected works: Series I.
     
   • Band 11 Teil 2 (German), [Bernard Bolzano - collected works: Series I.
     
   • Band 12 Teil 2 (German), [Bernard Bolzano - collected works: Series I.
     
   • Band 12 Teil 3 (German), [Bernard Bolzano - collected works: Series I.
     
   • Band 18 (German), [Bernard Bolzano - Collected works: Series 1.
     
   • Collected works: Series II.
     
   • Collected works: Series I.
     
   • Band 16 Teil 1 (German), [Bernard Bolzano - Collected works: Series I.
     
   • Band 16 Teil 2 (German), [Bernard Bolzano - Collected works: Series I.
     
   • Band 13 Teil 2 (German), [Bernard Bolzano - Collected works: Series I.
     
   • Band 4 Teil 1 (German), [Bernard Bolzano - Collected works: Series II.
     
   • (German) [Bernard Bolzano - Collected works: Series II.
     
   • Band 13 Teil 3 (German), [Bernard Bolzano - Collected works: Series I.
     
   • (German) [Bernard Bolzano - Collected works: Series II.
     
   • Band 5 Teil 2 (German), [Bernard Bolzano - Collected works: Series II.
     
   • Band 14 Teil 1 (German), [Bernard Bolzano - collected works: Series I.
     
   • Band 6 Teil 1 (German), [Bernard Bolzano - collected works: Series II.
     
   • 1827-1840 (German), [Bernard Bolzano - Collected works: Series II.
     
   • Band 6 Teil 2 (German), [Bernard Bolzano - collected works: Series II.
     
   • Band 7 Teil 1 (German), [Bernard Bolzano - Collected works: Series II.
     
   • Band 7 Teil 2 (German), [Bernard Bolzano - Collected works: Series II.
     
   • Band 8 Teil 1 (German) [Bernard Bolzano - Collected works: Series II.
     
   • Band 8 Teil 2 (German), [Bernard Bolzano - Collected works: Series II.
     
   • Band 14 Teil 2 (German), [Bernard Bolzano - Collected works: Series I.
     
   • Band 9 Teil 1 (German), [Bernard Bolzano - Collected works: Series II.
     
   • He also jots down his ongoing ideas on trigonometric series, the binomial theorem, Taylor's theorem, the mean-value theorem, and convergence of infinite series.
     
   • Band 14 Teil 3 (German), [Bernard Bolzano - collected works: Series I.
     
   • Band 10 Teil 1, Grossenlehre IV (German), [Bernard Bolzano - Collected works: Series II.
     
   • Band 9 Teil 2 (German), [Bernard Bolzano - Collected works: Series II.
     
   • Band 10 Teil 1 (German), [Bernard Bolzano - Collected works: Series II.
     
   • 1841-1847 (German), [Bernard Bolzano - Collected works: Series II.
     
   • Band 10 Teil 2 (German), [Bernard Bolzano - Collected works: Series II.
     
   • He is collecting material for his future mathematical texts and also shows a particular interest in trigonometric series.
     
   • Band 11 Teil 1 (German), [Bernard Bolzano - Collected works: Series II.
     

2. function concept
   
   • Euler allowed the algebraic operations in his analytic expressions to be used an infinite number of times, resulting in infinite series, infinite products, and infinite continued fractions.
     
   • He later suggests that a transcendental function should be studied by expanding it in a power series.
     
   • For example it had led him to define the gamma function and to solve the problem which had defeated mathematicians for some considerable time, namely summing the series .
     
   • Fourier showed that some discontinuous functions could be represented by what today we call a Fourier series.
     
   • Monthly 105 (3) (1998), 263-270.',18)" onmouseover="window.status='Click to see reference';return true">18] that confusion regarding functions had been due to a lack of understanding of the distinction between a "function" and its "representation", for example as a series of sines and cosines.
     
   • Fourier's work would lead eventually to the clarification of the function concept when in 1829 Dirichlet proved results concerning the convergence of Fourier series, thus clarifying the distinction between a function and its representation.
     
   • However, despite this, when he begins to prove theorems about expressing an arbitrary function as a Fourier series, he uses the fact that his arbitrary function is continuous in the modern sense! .
     
   • It therefore has a Taylor series which converges everywhere but only equals the function at 0.
     
   • In 1876 Paul du Bois-Reymond made the distinction between a function and its representation even clearer when he constructed a continuous function whose Fourier series diverges at a point.
     

3. Indian mathematics
   
   • An analysis of the weights discovered suggests that they belong to two series both being decimal in nature with each decimal number multiplied and divided by two, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500.
     
   • For instance, the development of number theory, the theory of indeterminates infinite series expressions for sine, cosine and tangent, computational mathematics, etc.
     
   • The most remarkable contribution from this period, however, was by Madhava who invented Taylor series and rigorous mathematical analysis in some inspired contributions.
     
   • These include: a formula for the ecliptic; the Newton-Gauss interpolation formula; the formula for the sum of an infinite series; Lhuilier's formula for the circumradius of a cyclic quadrilateral.
     
   • Of particular interest is the approximation to the value of π which was the first to be made using a series.
     
   • Madhava's result which gave a series for π, translated into the language of modern mathematics, reads .
     
   • abound with fluxional forms and series to be found in no work of foreign countries.
     

4. Calculus history
   
   • This is the first known example of the summation of an infinite series.
     
   • Barrow was in some way to blame for this since the publisher of Barrow's work had gone bankrupt and publishers were, after this, wary of publishing mathematical works! Newton's work on Analysis with infinite series was written in 1669 and circulated in manuscript.
     
   • Similarly his Method of fluxions and infinite series was written in 1671 and published in English translation in 1736.
     
   • In these two works Newton calculated the series expansion for sin x and cos x and the expansion for what was actually the exponential function, although this function was not established until Euler introduced the present notation ex.
     Go directly to this paragraph
     
   • You can see the series expansions for sine and for Taylor or Maclaurin series.
     Go directly to this paragraph
     
   • by the method of infinite series, .
     

5. Prime numbers
   
   • He was able to show that not only is the so-called Harmonic series ∑ (1/n) divergent, but the series .
     
   • The sum to n terms of the Harmonic series grows roughly like log(n), while the latter series diverges even more slowly like log[ log(n) ].
     
   • This means, for example, that summing the reciprocals of all the primes that have been listed, even by the most powerful computers, only gives a sum of about 4, but the series still diverges to ∞.
     

6. Pi history
   
   • In Gregory's series, for example, to get 4 decimal places correct we require the error to be less than 0.00005 = 1/20000, and so we need about 10000 terms of the series.
     
   • from which the first series results if we put x = 1.
     
   • and then calculate the two series obtained by putting first 1/2 and the 1/3 into (3).
     

7. Orbits
   
   • Although the motions of the planets were discussed by the Greeks they believed that the planets revolved round the Earth so are of little interest to us in this article although the method of epicycles is an early application of Fourier series.
     Go directly to this paragraph
     
   • He treated it as a restricted three body problem and used transformations to produce infinite series solutions for the longitude, latitude and parallax for the Moon.
     Go directly to this paragraph
     
   • The beginnings of his theory was published in 1847 and he had refined the theory until it was published in 2 volumes in 1860 and 1867 and was extremely accurate, its only drawback being the slow convergence of the infinite series.
     Go directly to this paragraph
     
   • He discussed convergence and uniform convergence of the series solutions discussed by earlier mathematicians and proved them not to be uniformly convergent.
     Go directly to this paragraph
     

8. Bolzano's manuscripts
   
   • Before the first volume in the series appeared in 1969 there were a number of related publications.
     
   • The first volume in the new series Bernard Bolzano-Gesamtausgabe published by Friedrich Frommann Verlag and edited by Eduard Winter, Jan Berg, Friedrich Kambartel, Jaromir Louzil, and Bob van Rootselaar, contains a biography of Bolzano together with details of the topics on which he worked: mathematics, logic, theology, philosophy and aesthetics.
     
   • The second volume, which set the scene for the whole series, appeared in 1972.
     
   • Further details of this series of volumes is given in .
     

9. Real numbers 2
   
   • This leads into the study of infinite series but without the necessary machinery to prove that these infinite series converged to a limit, he was never going to be able to progress much further in studying real numbers.
     
   • Among the forms of the completeness property he implicitly assumed are that a bounded monotone sequence converges to a limit and that the Cauchy criterion is a sufficient condition for the convergence of a series.
     

10. Fractal Geometry
    
    • sum of a convergent power series) would certainly produce such a curve.
      
    • In 1967, while still there, Mandelbrot wrote his landmark essay, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension [Science, New Series 156 3775 (May 5, 1967): 636-638.',8)">8], in which he linked the idea of previous mathematicians to the real world -- namely coastlines, which he claimed were "statistically self-similar".
      
    • He argued that [Science, New Series 156 3775 (May 5, 1967): 636-638.',8)">8] .
      

11. Sundials
    
    • Series A, Mathematical and Physical Sciences, 1974.
      
    • Series A, Mathematical and Physical Sciences, 1974.
      
    • If Menaechmus or someone else marked this path with a series of dots on a given day, he would 'discover' a hyperbola.
      

12. Chinese overview
    
    • He described multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures.
      
    • He also gave many results on sums of series.
      
    • He produced his own versions of logarithms, infinite series, and combinatorics which did not follow the style of western mathematics but his research naturally developed out of the foundations of Chinese mathematics.
      

13. Bourbaki 1
    
    • A large number of subcommittees were formed, given the size of the group, and these were to cover the following topics: algebra, analytic functions, integration theory, differential equations, existence theorems for differential equations, partial differential equations, differentials and differential forms, calculus of variations, special functions, geometry, Fourier series, and representations of functions.
      
    • He presented a series of theorems, all completely wrong, each attributed to a different fake mathematician.
      
    • Already in 1935 Bourbaki had taken the decision to produce a series of books which were linearly ordered in the sense that no reference could be made except to books earlier in the linear progression.
      

14. Topology history
    
    • The idea of connectivity was eventually put on a completely rigorous basis by Poincare in a series of papers Analysis situs in 1895.
      Go directly to this paragraph
      
    • Schmidt in 1907 examined the notion of convergence in sequence spaces, extending methods which Hilbert had used in his work on integral equations to generalise the idea of a Fourier series.
      Go directly to this paragraph
      

15. Pi history references
    
    • R Roy, The discovery of the series formula for π by Leibniz, Gregory and Nilakantha, Math.
      
    • I Tweddle, John Machin and Robert Simson on inverse-tangent series for π, Archive for History of Exact Sciences 42 (1) (1991), 1-14.
      

16. The number e
    
    • In 1668 Nicolaus Mercator published Logarithmotechnia which contains the series expansion of log(1+x).
      
    • The work involves the calculation of various exponential series and many results are achieved with term by term integration.
      

17. Measurement
    
    • An analysis of the weights discovered in excavations suggests that they had two different series, both decimal in nature, with each decimal number multiplied and divided by two.
      
    • The main series has ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500.
      

18. Ledermann interview
    
    • It was while he was at St Andrews that Walter suggested a cheap series of books for undergraduates.
      
    • His idea was taken up and the Oliver & Boyd series of mathematical texts was born.
      

19. Pi history references
    
    • R Roy, The discovery of the series formula for π by Leibniz, Gregory and Nilakantha, Math.
      
    • I Tweddle, John Machin and Robert Simson on inverse-tangent series for π, Archive for History of Exact Sciences 42 (1) (1991), 1-14.
      

20. EMS History
    
    • The Training Colleges provide Summer Courses for teachers in Infant and Junior Schools, and give courses in Rural Gardening, Country Dancing and - possibly - Elocution, but as for a course say on the best way of teaching logarithms to those who know no algebra, or the best way of teaching the convergence and divergence of series to those who have merely reached the standard of the leaving certificate, these problems are never attempted; perhaps because there is no one in the Training Colleges competent to deal with them, or perhaps because if someone did attempt to deal with them then he would have no audience.
      

21. Greek astronomy
    
    • Meton worked in Athens with another astronomer Euctemon, and they made a series of observations of the solstices (the points at which the sun is at greatest distance from the equator) in order to determine the length of the tropical year.
      

22. Golden ratio references
    
    • R Archibald, The golden section - Fibonacci series, Amer.
      

23. Indian mathematics references
    
    • K M Marar and C T Rajagopal, Gregory's series in the mathematical literature of Kerala, Math.
      

24. function concept references
    
    • G Ferraro, Some aspects of Euler's theory of series: inexplicable functions and the Euler-Maclaurin summation formula, Historia Math.
      

25. Water-clocks references
    
    • Series A, Mathematical and Physical Sciences, 1974.
      

26. Fractal Geometry references
    
    • Science, New Series 156 3775 (May 5, 1967): 636-638.
      

27. Sundials references
    
    • Series A, Mathematical and Physical Sciences, 1974.
      

28. Golden ratio references
    
    • R Archibald, The golden section - Fibonacci series, Amer.
      

29. Indian mathematics references
    
    • K M Marar and C T Rajagopal, Gregory's series in the mathematical literature of Kerala, Math.
      

30. function concept references
    
    • G Ferraro, Some aspects of Euler's theory of series: inexplicable functions and the Euler-Maclaurin summation formula, Historia Math.
      

31. Water-clocks references
    
    • Series A, Mathematical and Physical Sciences, 1974.
      

32. Fractal Geometry references
    
    • Science, New Series 156 3775 (May 5, 1967): 636-638.
      

33. Sundials references
    
    • Series A, Mathematical and Physical Sciences, 1974.
      

34. Set theory
    
    • Cantor moved from number theory to papers on trigonometric series.
      Go directly to this paragraph
      

35. Kepler's Laws
    
    • Kepler carried out the reduction to heliocentricity, and further simplifying procedures, in a series of steps: .
      

36. Coffee houses
    
    • These were not just impromptu lectures given in the course of discussion, but rather were properly advertised and usually not one off lectures but rather extended lecture series.
      

37. Pell's equation
    
    • To find the infinite series of solutions take the powers of 170 + 39√19.
      

38. Real numbers 1
    
    • He makes a series of definitions.
      

39. Fermat's last theorem
    
    • Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge, England the first on Monday 21 June, the second on Tuesday 22 June.
      Go directly to this paragraph
      

40. Elliptic functions
    
    • Both Wallis and Newton published an infinite series expansion for the arc length of the ellipse.
      Go directly to this paragraph
      

41. Egyptian mathematics
    
    • What is the quantity? Other problems involve geometric series such as Problem 64: divide 10 hekats of barley among 10 men so that each gets 1/8 of a hekat more than the one before.
      

42. Classical light
    
    • He used a rotating wheel with 720 teeth to break up a light beam into a series of pulses.
      

43. Indian numerals
    
    • The early use of such large numbers eventually led to the adoption of a series of names for successive powers of ten.
      

44. Poincaré - Inspector of mines
    
    • As one might imagine, Poincare's report is a remarkably carefully argued document where he details a whole series of possible causes and lists the evidence for and against each.