Jacques Hadamard, Hon.F.R.S.E.

by P H H Fantham

Jacques Hadamard was born on December 8, 1865, at Versailles, the son of the Latin teacher at the Lycée Louis le Grand. In 1892, at the height of his early achievements, he married Louise-Anne Trenel, who was a constant help to him throughout his long career; Hadamard never recovered from the shock of her death a few years ago. Tragedy indeed came early to his family when his two eldest sons, both army officers, were killed during the First World War; later, in the Second World War, his third son was killed. However, he is survived by his two daughters and the younger, Jacqueline, took her father's place on the Conseil de la Ligue des Droits de I'Homme on his retirement. Hadamard himself died on October 17, 1963, acclaimed the greatest mathematician of the time.

In his childhood, before taking an interest in mathematics, he followed his father's love of the classics and, in the Concours Général, an examination taken in all the high-schools in France, he was awarded a prize for Latin and Greek composition. It was only in the later stages at school, when working for the Baccalauréat that he changed his interests to mathematics. He was so successful that he was top of the entry list at l'école Normale Supérieure, where he continued his studies. About the same time, he also applied for entrance to l'école Polytechnique where he had the distinction of being the first person ever to gain full marks in the entrance examination.

His interests were not confined to mathematics. He was an ardent music lover and ran an amateur orchestra as well as playing in a string quartet. He was also keenly interested in botany and even professional botanists consulted him about ferns. An eminent chemist once said that he was quite amazed by Hadamard's knowledge of chemistry.

After taking his Aggrégation, he taught at the Lycée Buffon where he wrote his "Leçons de Géometrie Elementaire", a school text book of great originality. On submitting his thesis he was appointed first to the University of Bordeaux (1894-1897) and later to the Sorbonne. In 1912 he was elected to the titular chair at the Collège de France where he remained until his retirement in 1935.

His thesis, which appeared in 1892, dealt with analytic functions; it was here that he proved, among other things, his well-known formula expressing the radius of convergence of a power series in terms of its coefficients. The general idea of the thesis was to examine the properties of an analytic function, in particular, the singularities and properties of convergence, in terms of the coefficients; he then applied his results to the study of gapped series. During this early period he also examined series whose coefficients are the products of the coefficients of two given series, and at the same time investigated the maximum modulus of functions in a circular region thereby obtaining his celebrated three-circle theorem. Following up a problem posed by l'Académie in 189 1, he next worked on integral functions, expressing the genus and zeros of an integral function in terms of its coefficients. Hadamard's methods brought him fame when he was awarded the Grand Prix des Sciences Mathématiques in 1892. This work also brought him into contact with analytic number theory. He showed, for example, that the Riemann Zeta-Function has no zeros whose real part is unity, and he used this result to give a new proof of Cebicev's theorem on the distribution of prime numbers.

His curiosity for novelties soon led him further afield to the Calculus of Variations. Developing ideas from a course he gave at the Collège de France, he wrote a book on the subject in which he emphasized Volterra's technique of studying path-functions, that is, functions whose variable is a curve rather than a number. By analogy, he introduced what he called "functionals", that is, functions whose variable is a real-valued function. In his book, he systematically regarded the calculus of variations as a problem in the theory of functionals, or functional analysis, as we now call it. Later on, he adopted the definition of the derivative of a functional suggested by Volterra, and used this to study the effect on the Green's function of a domain, when the domain undergoes a continuous variation. In a prophetic article, he proposed the study of real-valued functions whose variable is of a quite arbitrary nature and which would necessitate, as he observed, the study of a new geometry. It was, in fact, this suggestion that led others to the study of abstract spaces.

From this period onwards, his studies in pure and applied mathematics became intimately connected. Very often, the problems involved had an interesting geometrical interpretation as, for example, when he discovered cases of discontinuity in apparently continuously formulated problems. Thus, for a particular type of surface, he showed that the set formed by the tangent lines is a disconnected set. This, and similar examples, arose out of his study of the integral curves of systems of differential equations which he used to elucidate problems of stability in mechanics. He introduced the so-called "fundamental solutions" of partial differential equations and was then able to give satisfactory answers to many problems for the first time. He also worked on fluid and wave mechanics; he was able to give a complete analysis of Huygens' principle and, again, he found examples of discontinuity, this time in the propagation of light.

In his last research period he returned to pure mathematics: he studied infinite integrals; he gave the first operator definition of the differential of a function, following up previous rigorous definitions by Stolz and Severi; he worked on quasi-analytic functions; on probability and, finally, produced an important theorem on determinants which was used in the classical formulation of Fredholm theory.

For his first few years at the Collège de France, Hadamard gave the traditional type of course. From 1913 onwards, however, he introduced a new procedure in which papers were analysed by members of the class; this was a forerunner of the modern seminar. He began the seminars himself by lecturing on the works of Henri Poincaré, for whom he had a great admiration, but thereafter, the students themselves lectured; both the papers and the students were chosen by himself. These seminars became so well-known that mathematicians came from all over the world to participate. Only Hadamard himself was capable of directing the discussions which led to so many different branches of mathematics, and he could always add new and profound remarks on any specialist topic.

In addition to his strictly scientific work in mathematics, he was also interested in the Philosophy of Science as is shown by his now famous correspondence attacking the empirical views of Borel, Lebesgue and Baire. During his stay in America he published a work on the psychology of invention. The scope of his interests was equalled by his wide travels. In addition to visiting Yale, he also visited Rio de Janeiro and he was for four months a professor at the Tsing-Hua University in Peking. His work brought him many honours. Having received numerous prizes from the Académie des Sciences, he was elected a member at the age of forty-seven. He was a Foreign Member of the Academia dei Lincei, of the Soviet Academy of Sciences, of the National Academy of Sciences and of the Royal Society of London. In 1946 he was elected an Honorary Fellow of the Royal Society of Edinburgh.

Fuller details of Professor Hadamard's life and work are to be found in the Notice Nécrologique by Professor M Fréchet, which was published by the Académie des Sciences of the Institut de France, December 23, 1963. The present obituary is based on Professor Fréchet's article.

Jacques Hadamard's RSE obituary by P H H Fantham appeared in Royal Society of Edinburgh Year Book 1965, 22-25.