**Georges Valiron**attended the lycée in Lyon. After graduating from the lycée he studied at the École Normale Supérieure in Paris, being the top student when he was awarded his degree in mathematical science in 1908. He then became a secondary school teacher of mathematics, spending four years teaching at Besançon. It was during these years that he taught André Bloch and his younger brother Georges. Valiron was awarded a bursary to enable him to study for his doctorate at the Faculty of Science in Paris and he spent the years 1912-14 undertaking research advised by Émile Borel. In fact he had begun undertaking research while still a teacher at Besançon and he had published

*Sur les fonctions entières d'ordre nul*Ⓣ in 1911. Further publications appeared:

*Sur les fonctions entières d'ordre nul et d'ordre fini et en particulier les fonctions à correspondance régulière*Ⓣ (1913);

*Sur quelques théorèmes de M Borel*Ⓣ (1914) and

*Sur le calcul approché de certaines fonctions entières*Ⓣ (1914). He defended his thesis

*Sur les fonctions entières d'ordre nul et d'ordre fini et en particulier les fonctions à correspondance régulière*Ⓣ before a panel at the Faculty of Science in Paris on 20 June 1914 and was awarded his doctorate. Let us note that although his thesis was not examined until June 1914, the 1913 publication with the same title is, in fact, his thesis and the 1914 papers were continuations of the research he presented in the thesis. This marks the start of his research career in which he made substantial progress in the study of the theory of entire functions, holomorphic functions, meromorphic functions and Tauberian theorems.

By the end of July 1914 France was mobilizing its troops and, on 3 August, Germany declared war on France. Valiron was mobilized and served in the artillery during the war. By the end of the war in 1918 he had risen to the rank of second lieutenant. He continued to serve in the military until he was demobbed in 1919. At that point he was appointed as a professor of mathematics at Valence, capital of the Drône department in southeast France, 100 km south of Lyon. He was sent to the University of Strasbourg to teach mathematics, presenting a course on 'Dirichlet series and factorial series' in March and April 1921. He taught for over ten years in Strasbourg and, at this point, we should explain the political significance of this move.

The Franco-Prussian War of 1870-71 had been a humiliation for the French. Germany had captured Strasbourg after a 50-day siege during the 1870-71 war and annexed the city. After taking control of Alsace-Lorraine, the Germans had reorganised the University of Strasbourg and reopened it as the German Kaiser-Wilhelm University of Strasbourg in 1872. Two mathematics chairs were founded at this time, the first was filled by Elwin Christoffel and the second was filled by Theodor Reye. Both Christoffel and Reye considered it their patriotic duty to assist in making the University of Strasbourg a German university. After the German defeat in World War I, Strasbourg reverted to French control in 1919 and the French re-established the University of Strasbourg, reorganising it as the autonomous French university Université Louis Pasteur-Strasbourg. Valiron was one of a number of eminent Frenchmen sent to Strasbourg to achieve this aim. He taught there until 1931, being promoted to professor in 1922. Maurice Fréchet was his colleague in Strasbourg from 1921 to 1927.

In February and March 1922 Valiron delivered a series of lectures to honours students at the University College of Wales at Aberystwyth. He had been invited to Aberystwyth by W H Young as part of a move to raise the status of Pure Mathematics in the University of Wales. The lectures were published in the following year as the book *Lectures on the general theory of integral functions*. Valiron delivered the lectures in French and they were translated into English by Edward Collingwood who, after undergraduate studies at Cambridge, had gone to Aberystwyth at the invitation of W H Young. In a Preface to the book, W H Young writes:-

Valiron writes in a separate Preface (which we have translated into English, but which appears in the book in French):-The lectures give us, in the form of a number of elegant and illuminating theorems, the latest word of mathematical science on the subject of 'Integral Functions'. And they do more. They descend to details, they take us into the workshop of the working mathematician, they explain to us the nature of his tools, and show us the way to use them; while, at the same time, by absence of any attempt to conceal the imperfections of the edifice so far constructed, they indicate to us the work still waiting to be done, they inspire us with desire and furnish us with the means of completing it ourselves.

The importance of this book is seen from the fact that it was reprinted in 1949 and continues to be in print with the latest edition appearing in 2007. Following the publication of this classic text, Valiron published a number of small books containing between 50 and 60 pages. The first of these wasThis book reproduces the lectures which I had the honour to give to the students of W H Young at the University College of Wales at Aberystwyth in February and March1922. May I be permitted to repeat here an expression of my gratitude to Professor Young who invited me to give these lectures, who has given me the means to publish them and who has kindly presented them to the public. If this book renders some service to students, their gratitude should go first to Professor Young who was the promoter.

*Fonctions Entières et Fonctions Méromorphes d'une Variable*Ⓣ (1925). K P Williams writes in a review [9]:-

In the following year he publishedSo brief a time has elapsed since the appearance of Valiron's former work, 'Lectures on the General Theory of Integral Functions', that one naturally expects the present small volume to follow essentially the development given in the former book. The condensation of material in order to confine the subject to fifty pages is most noticeable. On this account the beginner in this field will probably prefer the older book, where the pace is more leisurely and the exposition more detailed.

*Théorie Générale des Séries de Dirichlet*Ⓣ (1926). L L Smail writes in a review [7]:-

Among the papers that Valiron published in the years following World War I, we mention:The aim of the series to which this work belongs is to present in brief and compact form the most important results and outlines of methods in various mathematical subjects of current interest. This particular number on Dirichlet series gives an excellent survey of this interesting field which is now growing so rapidly. ... To those already interested in the subject and to those who wish know something of the results and methods without devoting too much time to it, this compact summary should prove very valuable.

*Les théorèmes généraux de M Borel dans la théorie des fonctions entières*Ⓣ (1920);

*Recherches sur le théorème de M Picard*Ⓣ (1921);

*Recherches sur le théorème de M Picard dans la théorie des fonctions entières*Ⓣ (1922);

*Sur les fonctions entières vérifiant une classe d'équations différentielles*Ⓣ (1923);

*Sur l'abscisse de convergence des séries de Dirichlet*Ⓣ (1924);

*Sur les surfaces qui admettent un plan tangent en chaque point*Ⓣ (1926); and

*Sur la distribution des valeurs des fonctions méromorphes*Ⓣ (1926). In 1931 he left Strasbourg when called to the Faculty of Science in Paris. In the following year he was a plenary speaker at the International Congress of Mathematicians held in Zürich from 5 September to 12 September 1932. At the Congress, he gave the lecture

*Le théorème de Borel-Julia dans la théorie des fonctions méromorphes*Ⓣ. In 1937 he published another small work, namely the 55-page tract

*Sur les valeurs exceptionnelles des fonctions méromorphes et de leurs derivees*Ⓣ. Mary Cartwright reviews this book in [2]:-

Joseph Walsh, reviewing the same tract, writes:-This tract is concerned with[a]theorem, which is due to Miranda, and with other similar results of a more complicated nature covering meromorphic as well as regular functions. The proofs are naturally very much condensed, and a good deal is assumed; but the references should be sufficient for anyone with a little knowledge of the general theory of integral functions. ... The later results in the tract are obtained by the author's direct method for proving Picard's theorem.

During World War II Valiron continued to teach in Paris. In October 1940 he took over the courseThe treatment is clear, pleasing, suggestive - an admirable exposition of a field of current interest and importance. May there soon be made in this country systematic provision for the encouragement of the writing of similar essays and for their publication!

*Application de l'analyse à la géométrie*Ⓣ which had been given by René Garnier. He was named professor of general mathematics in 1941 then, later in the same year, named professor of differential and integral calculus at the Faculty of Science. As well as teaching at the Faculty of Science, he taught at the École Polytechnique.

Valiron published his *Cours d'Analyse mathématique* Ⓣ in two volumes. The first, entitled *Theorie des fonctions* Ⓣ, was published in 1942. The contents are indicated by the titles of the chapters: I. Numbers, sets, limits; II. Infinite series and products; III. Generalities on continuous functions. Functions of a single variable; IV. The Riemann integral. Applications and extensions; V. The Lebesgue integral; VI. Functions of a single complex variable defined or represented by series or integrals; VII. Trigonometric series and generalizations; VIII. Reduction and mechanical computation of integrals; IX. Functions of several variables; X. Double integrals; XI. Triple and multiple integrals; XII. Elementary functions of a complex variable; XIII. The Cauchy theory. Fundamental theorems and the calculus of residues; XIV. The Weierstrass theory. Analytic continuation; XV. Conformal mapping; XVI. Elliptic functions; XVII. Analytic functions defined by integrals. Applications. Maurice Heins writes in a review:-

The second volume, published in 1945, was entitledNoteworthy features of the book are:(1)a careful detailed account of multiple integral theory under the usual classical assumptions;(2)a brief treatment of continued fractions, a topic not customarily treated in such texts;(3)a proof of the Hadamard theorem concerning the asymptotic behaviour of π(x), the number of primes not exceeding x;(4)contact with modern research in analysis by frequent reference to recent monographs dealing with special topics.

*Équations Fonctionnelles; Applications*Ⓣ>. It covers ordinary differential equations, partial differential equations, and algebraic equations of two variables. The author of [1] tries to put Valiron's treatise into the context of other classic texts on analysis:-

These two volumes were published as a single volume treatise in 1966, with another edition appearing in 1989.A new 'Cours d'Analyse' inevitably challenges comparison with the classic treatises familiar to every self-respecting student of mathematical analysis. The reviewer has always felt that of the four best-known classics, Goursat and de la Vallée Poussin exhibit a perfection of completeness and finish, while Jordan and Picard display the beauty and stimulus of a living and active organism. Such a contrast, if accurate, does not disparage either side ; and if we say that Valiron's book has more of the air of Goursat and de la Vallée Poussin, the implied mild regret is surely more than outweighed by the magnitude of the compliment. If, at times, the brilliant and lucid exposition of these authors leaves us feeling that nothing remains to be done, the brilliancy and lucidity still stand as well worth emulation.

Most biographies of Valiron state that his most famous doctoral student was Laurent Schwartz who was awarded his doctorate by the University of Strasbourg in 1943. This is slightly puzzling since Valiron spent the war years in Paris and had not taught at Strasbourg in the preceding ten years. Another point which requires clarification is the comment by one biographer that Valiron was not Laurent Schwartz's doctoral advisor, but rather an examiner of his thesis. He quotes Laurent Schwartz's autobiography as justification for this claim. Laurent Schwartz studied in Paris at the École Normale Supérieure in the 1930s and at this time took a course by Valiron on *Functions of a complex variable*. Schwartz then spent 1937-39 in military service which was extended for a year because of World War II. He was demobbed in August 1940 and went to live with his parents in Toulouse. He happened to meet Henri Cartan when he visited Toulouse to conduct an oral on behalf of the École Normale Supérieure and Henri Cartan advised him to study for a doctorate at Clermont-Ferrand which is where the University of Strasbourg moved when the German armies invaded France at the start of World War II. Laurent Schwartz's thesis contains the following acknowledgement as to the help that Valiron had given him:-

It is certainly true that Valiron was an examiner of the thesis, as Laurent Schwartz states in his autobiography. However, at another point in the autobiography, Schwartz writes that a Vietnamese student:-I want to especially thank Georges Valiron who not only gave me much advice, but also, through the correspondence he kindly entered into with me, helped me to overcome many difficulties.

There is, therefore, no doubt that Laurent Schwartz considered Valiron to have been his thesis advisor although he almost certainly did not have this as a formal role. We note also that Valiron examined the doctoral thesis of Laurent Schwartz's wife, Marie-Hélène, the daughter of Paul Lévy.... had(like me)Valiron as his thesis advisor.

Valiron received many honours. He was invited to lecture, or give plenary addresses at conferences, in many countries: England, Belgium, Switzerland, the United States, South America, and Egypt. He was elected president of the French Mathematical Society in 1938. He received five awards from the Academy of Sciences, including the Grand Prix of the Mathematical Sciences in 1931. He was awarded the Prix Poncelet in 1948. He was made Commander of the Légion d'Honneur in 1954. In the same year he published *Fonctions analytiques* Ⓣ. H L Royden writes [6]:-

Maurice Heins, reviewing the same book, writes:-This book is intended for the student who has had an introductory course in functions of a complex variable. ... Altogether this is a very useful little book which has collected together a number of diverse topics in function theory in a form which is easy to read and quite suitable for the student who wishes to familiarize himself with some of the classical tools available in function theory.

The monograph under review is concerned with selected topics in the theory of analytic functions. ... autonomy of exposition is sought as well as the avoidance of duplication with other well-known monographs on the theory of functions. Emphasis is placed on expounding the more elementary parts of the theories studied. An abundance of references points the way to further detailed study of the topics treated. The geometric approach is stressed. ...[A]wealth of material[is]considered. The exposition is lucid. This book should be of considerable value for continuing courses in the theory of analytic functions.

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

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