Died: 21 January 1974 in Paris, France

**Arnaud Denjoy**'s father, Jean Denjoy, was a wine merchant in Perpignan, which is in southern France about 30 km north of the border with Spain. We do not know Arnaud's mother's first name, but her surname was Jayez and she came from Catalonia. Arnaud was born and attended a secondary school in Auch which is the capital town of the Gers region in southwestern France. After attending secondary school in Auch, Arnaud completed his schooling in Montpellier which is close to the Mediterranean coast in southern France. He remained associated with these regions throughout his life.

In 1902 Denjoy entered the École Normale Supérieure where he studied under Borel, Painlevé and Émile Picard. These great mathematicians gave Denjoy a strong background in complex function theory, continued fractions and differential equations and set him on the road to his great discoveries. Denjoy enjoyed the highest success during his undergraduate years, being the top student in his class. His success was translated into the winning of a prestigious fellowship, the Fondation Thiers fellowship, which supported him for three years during which he worked on his dissertation *Sur les produits canoniques d'ordre infini* which he submitted in 1909.

Denjoy's dissertation, although not considered by him as among his greatest achievements when he looked back on his career in 1934, is now considered to contain some remarkable contributions. It studies the asymptotic behaviour of integral functions of finite order, Weierstrass products of integral functions and the boundary behaviour of conformal representations.

From Paris, Denjoy moved back to Montpellier in 1909 when he was appointed "Maitre des conferences" at the University of Montpellier. This post saw him preparing students for the "agregation" examination and it was the same post in the same university to which Baire had been appointed in 1901. Baire had left Montpellier before Denjoy took up this appointment, but he was an important figure in the development of Denjoy's mathematics. Denjoy was one of a small number of people who appreciated Baire's innovative ideas when he first produced them. For example when Baire was told that what he taught was so difficult that it was beyond human ability to understand it, he wrote:-

... but look at Denjoy - he understood it, hence it must not be so difficult...

Denjoy taught at Montpellier until the start of World War I in 1914. He suffered from poor eyesight so he was not fit for military service during the war. He was appointed to a professorship at Utrecht in 1917 then at the University of Paris from 1922, a post he held until he retired in 1955. Not long after his appointment at the University of Paris, he married Thérèse-Marie Chevresson in June 1923. They had three children, all sons.

Denjoy worked on functions of a real variable in the same areas as Borel, Baire and Lebesgue. He combined topological and metrical methods to attack problems of real analysis. 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. Included in these papers was his introduction of the Denjoy index for the points of a perfect set. Also in these papers is his introduction of the second symmetric derivative of a function. This work is studied in detail in [3] where Bullen makes the contents of Denjoy's work, which is not easy to read, more accessible to modern analysts.

Although considered by Denjoy as one of his most important pieces of work in his 1934 review, Choquet writes in [1] that these papers may be:-

... considered more feats of intellectual strength than sources of practical applications.

However, Choquet describes the four volume work *Lectures on the computation of coefficients in a trigonometric series* which contains the famous Denjoy integral, as [1]:-

... an explosion of beautiful theorems and examples.

Choquet, very fairly, suggests that Denjoy's work on differential equations on a torus, not nearly so highly rated by Denjoy himself, is one of his most influential pieces of work and has [1]:-

... grown into a vast field involving dynamical systems.

Similarly Denjoy's theorem on quasi-analytic functions has been the foundation of studies by Mandelbrojt and has proved important in the development of large areas of current research.

Let us end by saying something of Denjoy's character. He was a quiet man, who liked to carry out his mathematical research in the peace of his own home. For leisure he enjoyed being in the countryside, particularly walking and cycling in wooded country. A relatively poor lecturer he was, nevertheless, a fine writer and an entertaining man with whom to hold a conversation.

Denjoy was not a man lacking interests outside mathematics: on the contrary he was fascinated by topics such as philosophy, psychology, and social studies. He approached such topics from his position as an atheist and an active participant in socialist politics. In particular he was an active member of the Radical Party which was headed by Édouard Herriot. Although Denjoy did not aspire to the political career of Herriot, who served in nine different cabinets and was premier of France three times, Denjoy's involvement with the Radical Party led to him serving as a town councillor for Montpellier in 1912, and as county councillor for Gers from 1920. He served in this capacity for twenty years.

For his outstanding contributions to the theory of functions of a real variable, Denjoy received many honours. As well as election to the Académie des Sciences in 1941, he was also elected to the academies in Amsterdam, Warsaw, and Liege. He was also honoured by being elected as vice-president of the International Mathematical Union in 1954.

**Article by:** *J J O'Connor* and *E F Robertson*

**July 2000**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Denjoy.html]