Jacques Hadamard Jean-Pierre Kahane

In introducing Jacques H a d a m a r d to the London Mathematical Society in 1944, G. H. Hardy called him the "living legend" in mathematics. Legendary work, flourishing over function theory, number theory, geometry, mechanics, ordinary and partial differential equations. Legendary figure, running the first mathematical seminar in the Coll6ge de France, travelling over all continents, involved in actions for human rights and world peace, celebrated among mathematicians and ordinary people, a little man with a great character, le petit p~re Hadamard, as we called him affectionately in the 1950s. Nevertheless, no mathematical library contains the whole mathematical work of Hadamard, because--unlike the work of much less important mathematicians--it was never collected and published in its entirety. No street in Paris wears his name. The l e g e n d n e e d s a revival, especially in France. The life of Jacques Hadamard extends over almost a century: 8 December 1865 to 17 October 1963; about the same period of time as David Hilbert (1862-1943). Henri Poincar6 (1854-1912) was more than ten years older, Emile Borel (1871-1956), Ren6 Baire (18741932) and Henri Lebesgue (1875-1941) younger. The turn of the century was extremely bright for mathematics in France. Without a doubt, Poincar6 was le prince des math~maticiens." In the younger generation Hadamard played a leading role, and I shall discuss it briefly in a moment apropos of set theory. Hadamard's life spans three wars on French soft, six political regimes in France, from Napoleon III to De Gaulle, and such events as the Paris Commune, the Dreyfus affair, the Russian Revolution, the rise of fascism in Europe, the Front Populaire, the shame of Vichy, the Resistance, Hiroshima, the cold war, the

first man in space. He was involved as a citizen in many aspects of our history. I shall divide this paper into two parts. In the first I shall try to evoke the long life of Hadamard and his involvement in events of the time without saying much of his mathematical work. By selecting a few topics or notions to which his name is attached, in the second part I intend to give an idea of the variety and the richness of his mathematical production.

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children, and looked a most marvellous couple up to her death in 1960 after a marriage that lasted sixtyeight years), and obtained the Grand Prix des Sciences Math4matiques for his work on entire functions and in particular on the function ~ considered by Riemann: ~(t) = F

+ 1 (s - 1)'rr-s/2~(s)

1

(s = -2 + it)

where he proves that ~(t) is of the form ~(t) = C [ I

Jacques and Louise-Anna Hadamard on a visit to India.

Life and Time

Jacques Hadamard originated from a Jewish family of Lorraine. There are traces of Hadamards, printers in Metz, in the eighteenth century, and also of a very remarkable great grandmother who lived during the French Revolution. Before Jacques was born the family settled in the Paris area. His father taught humanities in high school, his mother was a good pianist. The war with Prussia, the defeat, the fall of Napoleon III occurred w h e n he was not yet five years old. What he remembered later from the siege of Paris (winter of 1870) was eating a piece of trunk of the elephant of the Jardin des Plantes. At school he was bright in every subject but mathematics. "To parents in despair because their children are unable to master the first problems in arithmetic I can dedicate m y example. For, in arithmetic, until the seventh grade, I was last or nearly last," he said in 1936. ("Je puis dddier mon exemple aux parents que ddsesp~re l'inaptitude de leurs enfants ~ triompher des premiers probl~mes d'arithmdtique; car en arithm~tique--jusque et y compris la cinqui~me--j'~tais le dernier ou d~peu pros".) He was particularly successful in Latin and Greek. Howe v e r - - d u e to an inspiring t e a c h e r - - h e discovered the beauty of mathematics, moved to science, prepared for the competitive entrance exams of Ecole Polytechnique and Ecole Normale Sup6rieure, and got first place in both, with the highest score ever seen at the Ecole Polytechnique. He chose Ecole Normale Sup6rieure (1884), where he studied under Jules Tannery ("the scientific guide") and Emile Picard ("the superb teacher"). After leaving Ecole Normale Sup6rieure he taught as a high school teacher (a pupil of his was Maurice Fr6chet) and prepared his thesis. In one year (1892), he got his doctorate (on functions defined by Taylor series), married his beloved mate Louise-Anna Trenel (a beautiful romance: they loved each other, he waited too long, she got engaged to another man, he jumped, pleaded, succeeded, they married, had five 24

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,

The story deserves telling. The Acad6mie des Sciences had proposed a study of ~r(x), the number of prime numbers less than x, and everybody expected the Prize to be given to Stieltjes, who had just published a note saying that he had proved the Riemann hypothesis. However, the proof did not come, ~(t) has much to do with ~(x), and Hadamard got the Prize. Four years later, he actually proved the prime number theorem: "rr(x) - ei x. There are many anecdotes about Hadamard as a young man. According to his daughter Jacqueline, he was the model for le savant Cosinus, a kind of inteUectual comic of the early 1900s. Le savant Cosinus is incredibly forgetful in practical matters. Actually this was the case for Hadamard. Here is an example. He liked to pick herbs, and once took his little sister on an herb-picking expedition in the Alps; he settled her on the edge of a glacier, went on collecting leaves, went back home, and only then discovered that he had left the child behind in a dangerous situation. Another, even more dramatic example occurred much later, in 1940, w h e n he succeeded in leaving France for the United States, but left behind the briefcase containing the American visas. The years 1892-1912 were extremely fruitful from a mathematical point of view. After Bordeaux, Hadamard obtained a chair in Paris. In 1909, he was elected at the Coll~ge de France and in 1912 at the Acad4mie des Sciences. After 1909 the famous Hadamard seminar began at the Coll~ge de France, where every year he selected topics and speakers from the whole mathematical world. It was a happy time at home: bright children, music, good friends like Borel, Lebesgue, the physicists Paul Langevin and Jean Perrin and their German friend Albert Einstein, who played violin with Hadamard. However, it was not possible to ignore the external world. Captain Alfred Dreyfus, a relative of Hadamard, attached to the Deuxi~me Bureau of the French Etat-Major, had been accused of spying, was judged, condemned, and deported in rather strange circumstances, with no proof and a heavy climate of antiSemitism (1894). Hadamard did not feel involved at

the beginning. But the truth was revealed fact by fact: Dreyfus was innocent, another officer was guilty. France was divided into dreyfusards (Zola, Clemenceau, Hadamard) and antidreyfusards (most dignitaries in the Army and the Catholic Church). Only in 1906 was Captain Dreyfus reinstated in his rank and civil rights. Meanwhile, Hadamard took an active part in the Dreyfusan Ligue des droits de l'homme, in which he remained a member of the Central Committee until late in life and then was replaced by his daughter Jacqueline. L'Affaire Dreyfus played an important role in France and in Hadamard's life. As an example of the climate of the time, Charles Hermite once, crossing Hadamard, shouts: "'Hadamard, vous ~tes un traitre'" (you are a traitor). Then, before Hadamard can react, " V o u s avez trahi l' analyse pour la gdorn~trie" (you betrayed analysis for geometry). A typical bad joke of the time, when treason and treachery were key words of the social life in France. A tragedy of another magnitude began in 1914. Pierre and Etienne, the elder sons of Hadamard, were killed in 1916, and the obituaries written by Jacques Hadamard show that he was struck in a terrible way. Almost all the students of Ecole Polytechnique and Ecole Normale Sup6rieure were killed too. The First World War was a scientific disaster for France. Between 1918 and 1939 Hadamard was oriented to the left, mainly due to the rise of Nazism in Germany and its copies in France. He was a convinced antifascist. In 1938 he was one of the few in France--apart from c o m m u n i s t s - - t o be indignant towards the Munich a g r e e m e n t b e t w e e n Chamberlain, Daladier, Hitler, and Mussolini about Czechoslovakia. Here is a letter the H a d a m a r d s sent to their colleagues in Prague (copy due to Dr. Vladimir Korinek and Jacqueline Hadamard): Dear friends, We need not tell you how close we feel to you in these days of mourning. You at least do not wear any shame and can be proud of maintaining your honour untouched. The behaviour of your President, his constant dignity, are and will be admired all around the world, and hopefully will be retained by immanent justice. Above the Western governments who betrayed you and betrayed us we shake hands with you. Jacques and Louise Hadamard. (Mes chers amzs, En ces jours de deuil, est-zl besoin de vous dire comhen nous sommes proches de vous. Du moins n'avez-vous pas la honte et pouvez-vous avolr la fiert~ de dire que vous avez maintenu haut votre honneur. L'attitude de votre Pr~szdent, la &gnit~ dont il ne s'est pas ddpart~ un instant, ont fait l'admiraflon de tous et feront celle de l'Histoire, et, on est en droit de l'esp~rer, la justzce immanente s'en souviendra. Par dessus les gouvernements occzdentaux qui vous ont trahzs et nous ont trahis nous vous serrons la main.)

After the French defeat in 1940, Jacques, Louise, and Jacqueline got their American visa and escaped the racial persecutions, thanks to the Joint Jewish Committee and to a very active and efficient young

Jacques Hadamard with his daughter Jacqueline.

Canadian scientist, Louis R a p k i n e - - I already mentioned the tragicomic incident of the lost briefcase. They settled in N e w York. Hadamard lectured at Columbia University and wrote his beautiful book on the psychology of invention in mathematics. A last personal tragedy was the death of Mathieu, his last son, officer in the Free French Forces, in 1942. When he returned to Paris in 1944, Hadamard had no apartment left, none of the books and papers that he had left behind, and had to begin a new life. He got involved more and more in social and political questions. His daughter Jacqueline joined the Communist Party. He became very active in the Peace Movement, then led by Fr~d6ric Joliot-Curie, and he was sometimes considered a communist. This happened to have an effect on the world scene. In 1950, the first International Congress of Mathematicians after the war was held in Cambridge, Massachusetts, and Hadamard was elected honorary president of the Congress. However, it was the time of cold war and McCarthyism, and at first Hadamard was denied an American entrance visa, as was Laurent Schwartz, who was to receive the Fields Medal. A few French mathematicians would have agreed to attend the Congress anyhow, but most said they would not, and the American mathematical community was strong e n o u g h to convince the American g o v e r n m e n t to change its position. THE MATHEMATICAL INTELLIGENCER VOL 13, N O 1, 1991 2 5

Hadamard had been involved in mathematical education since his youth. In 1932 he became president of the Commission Internationale de l'Enseignement Math4matique, which had been created by Felix Klein, George Greenhill, and Henri Fehr in 1908. In 1936 he retired from Coll~ge de France. From his scientific Jubil6 in 1936 until his death in 1963, Hadamard received many more h o n o u r s than I could mention in a few pages. He was particularly pleased when he received the FeltrineUi Prize, which had just been created in 1955 in order to replace the Nobel Prize in mathematics; the speech was made by Vito Volterra, the Prize was given by the President of the Republic--it was really a great event in Italy--and Hadamard was angry only about the French ambassador w h o did not find it convenient to attend the ceremony (maybe such anger was part of the pleasure!). Before coming to the second part, let me thank Jacqueline Hadamard* for her help; most unpublished anecdotes are taken from a personal manuscript of hers. Some others are recollected from talks by my teacher, Szolem Mandelbrojt--I am just sorry that I am not up to his warmth and wit. I personally had a few occasions to meet Hadamard in the early 1950s, once at his home, rue Emile Faguet, in one of these university apartments near the Citd Universitaire which were built on his initiative. Otherwise I saw him in different meetings. He was always late, always tiptoed to the floor, asked for a chair, and let his fingers drum until he was invited to say a few words. What I remember best is the expression of his face. From the photographs you can appreciate its sharp and biblical beauty, but more striking was the acuteness of his look and the constant motion of his eyes. More than eighty-five years old, he was not only a living legend, but a truly living mind. Let us come to what remains of him, his written work.

Pearls and Threads There are a b o u t 300 scientific p a p e r s and books written by Jacques Hadamard. I shall select a very few of the results and notions due to him.

The prime number theorem: ~r(x) - r x (1896). It was an old conjecture, known to be derived from the Riemann hypothesis. Hadamard shows that it follows from the nonvanishing of ~(1 + it), and actually proves ~(1 + it) ~ O. This was proved at the same time and independently by Charles de Ia ValiSe Poussin. Hadamard's proof that ~(1 + it) ~ 0 is beautiful. He considers log ~(s) = Xann -s and uses the fact that an I> 0. Since ~(s) has a pole at s = 1, log ~(1 + e) = log 1/e + 0(1) as 9 ~, 0. If ~(s) had a zero at s = 1 + ia, then *Jacquehne Hadamard died shortly after flus article was completed. 26

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log ~(1 + ia + e) = - log lie + 0(1), therefore (using an i> 0) n -~ -~ - 1 for most n's, therefore n -2za ~ 1 for most n's, therefore log ~(1 + 2ia + e) ~- log l/e, impossible since 1 + 2/a is a regular point for ~(s). The upper limit is used in the formula R = (lim sup Ic2,n) -,, giving the radius of convergence R of a Taylor series ECnzn (1892). Though the formula can be found in an 1821 monograph of Cauchy, Hadamard discovered it independently and drew far-reaching conclusions from it. It provided a wealth of results on the location of singular points and was the starting point of a deep investigation of the global behaviour of a function from its Taylor coefficients. Hadamard's booklet on Taylor series and analytic continuation (La s~rie de Taylor et son prolongement analytique, 1901) was the source of most of the 300 papers quoted in Bieberbach's Analytische Fortsetzung (1954).

In 1950, the first International Congress of Mathematicians after the war was held in Cambridge, Massachusetts, and Hadamard was elected honorary president of the Congress. The multiplication theorem (1898) is a pearl of the theory of functions. Roughly speaking, it states that the singular points of E~ anbnz n are contained in the product set AB (= {ot~:ct ~ A, f~ (B}), where A and B are the sets of singular points of ~ anzn and ~ bnzn, respectively. It may remind us of another theorem: the support of a convolution product is contained in the sum of the supports of the factors. Both statements are very similar (and need to be made precise). Hadamard's multiplication theorem is actually an early and excellent e x a m p l e of the p o w e r of c o n v o l u t i o n m e t h o d s - - b e f o r e the notion of convolution was born.

The Hadamard lacunarity condition h n+ l/hn > q > 1 appeared also in connection with analytic continuation. If this condition holds the series E~ anZXn is not continuable across the circle of convergence (1892). This is the prototype of a series of statements of the

type: if a function with a given spectrum enjoys a property on an interval and if the interval is larger than some density of the spectrum, then the property holds everywhere (here, the density is zero and the property is that of being analytic). P61ya, Mandelbrojt, Paley and Wiener, Ingham, and many others contributed theorems of this type. The three circles theorem states that log M(e~) is a convex function of or, where M(r) = suplzl= r If(z)l, f(z) being an analytic function in an annulus r 0 < Iz[ < r 1. Expressed for a strip instead of an annulus, it plays a fundamental role in interpolation theory (theorem of Riesz-Thorin, and the complex method of interpolation).

"Almost convexity" of log M n, where M, = supx [fn(x)l, f bein_g a C| function on ~. Hadamard's theorem is M1 V~0/~, a precise estimate (1914). Precise constants for other inequalities of this type were obtained by A. Kolmogorov, Szolem Mandelbrojt, and Henri Cartan. The problem of quasi-analyticity of a class C(M,) (C| functions g such that Ig(')(x)l ~ M, on a given interval) consists in deciding whether or not the functions in C(Mn) are defined by their germs at a given point. Hadamard asked the question in relation to boundary values of solutions of partial differential equations (1912). The solution was guessed and partially proved by Denjoy, proved completely by Carleman. The linkage between quasi-analyticity and spectral properties was introduced by Mandelbrojt. Here is the general problem of Mandelbrojt: given conditions S on the spectrum, given a class of functions C (such as L1, C, C(M,)" 9 "), consider functions of the class C satisfying S; suppose we know a property of such a function on an interval, or in the neighborhood of a point, or at a point; to what extent does it give information on the whole function? This kind of question appears again and again in partial differential equations, in particular now in control theory. The real part theorem, in its simplest form, says that M(r) ~ 2A(2r), where M(r) = suPlzl= r If(z)l, A(r) = suplzl= r Re f(z), f(z) being an analytic function in Iz < R vanishing at 0, and 2r < R (1892). It plays a key role in the factorization of entire functions.

N o w let m e e x p l a i n h o w H a d a m a r d t u r n e d "traitor," as Hermite said, betraying analysis for geometry. I suppose that Hermite did not have in view the book Lefons de gdorndtrie dldmentaire (1898), which proved quite influential among high school teachers, but Hadamard's works on geodesics, trajectories of differential equations, and analysis situs in the sense of Poincar6 (1896-1910).

Jacques Hadamard Geodesics on surfaces are a beautiful subject and the essential facts were discovered by Hadamard. On surfaces with positive curvature every nonclosed geodesic cuts every closed geodesic infinitely m a n y times (1896). On surfaces with negative curvature any two geodesics have at most one point in common (this was known before Hadamard); given any arc, there exists a unique geodesic arc with the same end points in the same homotopy class. Now comes the most important result, about the asymptotic behaviour. There are four cases: 0) closed geodesics; 1) geodesics tending to infinity; 2) geodesics tending asymptotically to a closed geodesic; 3) erratic geodesics visiting asymptotically neighbourhoods of different closed geodesics. In this last case the sequence of neighbourhoods to visit is quite arbitrary. And Hadamard studies how, given a point P, a geodesic starting from P behaves according to its initial direction 0. Here is a striking use of the notions introduced by Cantor: the set of 0 corresponding to bounded geodesics is perfect and totally disconnected. Therefore the boundedness property is not preserved by an infinitesimal variation of 0. Here is a comment of Hadamard:

It may be that one of the fundamental problems of celestial mechanics, the stability of the solar system, belongs to the category of ill-posed problems. If actually, instead of looking for the stability of the solar system, we consider the analogous question related to geodesics of surfaces with negative curvature, we see that each stable trajectory can be transformed into a totally unstable trajectory going to infinity through an infinitely small change in initial data. Since initial data in astronomical problems are THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

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known only within a small error, such an error might lead to an absolute perturbation in the result. (Un des probl~mes fondamentaux de la M~camque c~leste, celuz de la stabzht~ du syste~ne solaire, rentre peut-$tre dans la catdgorie des questions mal pos~es. Sz, en effet, on substztue ~ la recherche de la stabiht~ du syst~me solaire la questzon analogue relative aux g~od~szques d'une surface ~ courbure n~gative, on voit que toute trajectoire stable peut ~tre transform~e, par un changement infiniment petzt dans les donn~es mitiales, dans une trajectoire compl~tement instable se perdant ~ l'infini. Or, dans les probl~mes astronomiques, les donndes initiales ne sont ]amms connues qu'avec une certaine erreur. Si petite solt-elle, cette erreur pourrmt amener une perturbation totale et absolue dans le r~sultat cherch~.)

Partial differential equations was a favourite subject of Hadamard, from 1900 until very late in life. The basic book Lefons sur la propagation des ondes et les dquations de l'hydrodynamique appeared in 1903. He considered the Dirichlet problem (one boundary datum at the boundary) and the Cauchy problem (two data on the initial subspace t = 0); for an elliptic operator, the Dirichlet problem is well posed, and for a hyperbolic operator the Cauchy problem is well posed. Well posed in the sense of Hadamard is still a term in use in P. D. E.'s. Hadamard claimed that a wellposed problem is not only one for which the solution exists and is unique for given data; he insisted that the solution should d e p e n d continuously on the data; only such a solution has a physical meaning. In order to elaborate the notion he introduced different types of neighbourhood and continuity. This led to functional spaces, general topology, functional analysis, and to the a priori m e t h o d s u s e d in the modern theory of P. D. E.'s. It is worth noting that the term functional (fonctionnelle) was introduced by Hadamard, inspired by Volterra's fonctions de lignes, and that he gave a general expression for the linear functionals on the class of continuous functions on an interval (equivalent of course to the F. Riesz theorem, but not so simple to write and to use).

Elementary solutions (solutions ~l~mentaires, sometimes translated as fundamental solutions) were also introduced by Hadamard, in a sense slightly different from Laurent Schwartz's, simply because Schwartz's distributions did not exist yet. Clearly, a good part of

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THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

Schwartz's inspiration came from his grand-uncle Hadamard. In the theory of distributions one of the most elaborate notions is finite part (partie finie) of a divergent integral, and it was actually introduced and developed by Hadamard (1908). It is a beautiful and simple calculus of divergent integrals of the form f~x-~f(x)dx (or > 1 and ct non-integer) or generally fv(G(x))-~f(x)dx (x = (xl, 9 "", xn), V included in the hypersurface G = 0). Using this as a tool for the Cauchy problem, Hadamard is able to solve the problem when n is odd, and he introduces the descent method (m~thode de descente) when n is even, solving the problem for n + I and then going d o w n to n. P. D. E.'s were the subject of Hadamard's lectures at Yale University (1920). They resulted in a very inspiring book on the Cauchy problem and hyperbolic P. D. E.'s (English version 1922, French enlarged version 1932). Speaking on Hadamard's work in a ceremony held in 1966 on the occasion of his centenary, Laurent Schwartz expressed a general feeling in saying that Hadamard had a fantastic influence on his time, and that all living analysts were shaped by him, directly or indirectly ("Je crois qu'il a eu une influence ~norme sur son temps, que tousles analystes d"aujourd'hui ont ~t~, directement ou indirectement, formds par Jacques Hadamard."). The few examples I gave are just a very lacunary sample of his mathematical production. I did not enter the realm of mechanics and calculus of variations. Let me go on with three more topics: determinants, set theory, philosophy of mathematics.

The Hadamard inequality on determinants states that a determinant is dominated by the product of the euclidean norms of its columns. In the short paper he wrote on this topic (1893) Hadamard considers determinants whose entries are + 1 or - 1 , and the case when the bound n n/2 is attained; then (except for n = 1 or 2) it is necessary that n = 0 (mod 4) and Hadamard constructs examples for n -- 2k, n = 12, n = 20. Such examples (now obtained up to n = 264) are known as "Hadamard determinants," and they happen to play a role in the theory of error-correcting codes. Hadamard's contribution to the First International Congress of Mathematicians (Ziirich 1897) was on some possible applications of set theory (sur certaines applications possibles de la th~orie des ensembles). Read now, this paper introduces the notion of e-entropy of Kolmogorov; but it was too much in advance and forgotten. In 1905 the Bulletin of the French Mathematical Society published "Five Letters on Set Theory," an exchange of letters b e t w e e n H a d a m a r d , Borel, Lebesgue, and Baire. Hadamard appears as the leading force, not only for carrying out such a correspondence, but for advocating the free use of powerful

methods introduced only a short time earlier (what we now call Zermelo's axiom of choice). I already mentioned his use of Cantor sets in connection with the classification of geodesics. Hadamard's philosophy in mathematics was called idealistic at the time; this means only in opposition to the more constructive approach of Borel. He was influenced by Poincar6 and he has been most influential himself in heuristics. His book on The Psychology of Invention in the Mathematzcal Field (1945) was a constant reference for George P61ya, in particular. This book is an example of what Hadamard was able to write for a general audience. He knew how to speak or write on mathematics, not only in mathematics. He had a very broad view of mathematics. He was able to develop the most abstract parts and at the same time to be inspired by considerations from physics. I already mentioned that he was involved in mathematical education. His book on elementary space geometry has just been published anew (a good sign for the revival of geometry in mathematical training). His papers on scientific education are worth reading: he was more interested in the teaching of experimental sciences than in the teaching of mathematics (again, something to be considered now). Unfortunately, not many of them can be consulted easily. I am sure that the present mini-review will provide the reader with a feeling of frustration. I shall be happy if this frustration leads readers to more substantial articles (I give some in the references) and more than happy if readers go to the inspiring source, the articles and books of Jacques Hadamard.

Jacques Hadamard and E. G. Kogbetliantz at the International Congress of Mathematicians, Harvard, 1950.

8. Szolem Mandelbrojt and Laurent Schwartz, Jacques Hadamard (1865-1963), Bull. Amer. Math. Soc. 71 (1965), 107-129. 9. Francesco Giacomo Tricomi, Commemorazione del Socio straniero Jacques Hadamard, Atti della Accad. Nat. dez Lincei, Rendzcontz classe di sczenze fisiche, matematzche e naturah, 39, 5 (1965), 375-379. By Hadamard

References

On Hadamard 1. M. L. Cartwright, Jacques Hadamard, J. London Math. Soc. 40 (1965), 722-748. 2. Centenaire de Jacques Hadamard, Math6maticien (1865-1963), La jaune et la rouge, n ~ 204, mai 1966. (Ecole Polytechnique). 3. Maurice Fr6chet, Notice n6crologique sur Jacques Hadamard, prononc6e le 23 decembre 1963 devant l'Acad6mie des Sciences (r6sum6: CRAS Paris, tome 257, 4081-4086; version integrale: "Notices et Discours" publi6s par l'Acad6mie des Sciences, Gauthier-Villars 1963). 4. Paul L~vy, Jacques Hadamard, sa vie et son oeuvre. Calcul fonctionnel et questions diverses, L'Enseignement Math~matzque (2)13 (1967), 1-24. 5. Bernard Malgrange, Equations aux d6riv6es partielles, L'Enseignement Math~matique (2)13 (1967), 35-48. 6. Paul Malliavin, Quelques aspects de l'oeuvre de Jacques Hadamard en g6om6trie. L'Enseignement Mathdmatique 13(2~me sdrze), (1967), 49-52. 7. Szolem Mandelbrojt, Th6orie des fonctions et th6orie des hombres dans l'oeuvre de Jacques Hadamard, L'Ensezgnement Math~matzque (2)13 (1967), 25-34.

1. Oeuvres de Jacques Hadamard, 4 volumes. Paris: CNRS (1968). 2. Lemons de gdom~trie ~l~mentazre. Vol. 1: g~om~trze plane. Paris: A. Colin (1898). Vol. 2: gdomdtne dans l'espace. Paris: A. Colin (1901). (Reprinted by J. Gabay, Paris, 1988). 3. La s~rie de Taylor et son prolongement analytzque. Paris, Scientia, 1901. Deuxi~me 6dition compl6t6e, en collaboration avec Sz. Mandelbrojt. Paris: Gauthier-ViUars (1926). 4. Lefons sur le calcul des vamations. Paris: Hermann (1910). 5. Lectures on Cauchy's problem m hnear partial differentzal equations. Cambridge-New Haven (1922). (Traduction fran~aise: Paris: Hermann (1932)). 6. Cours d'analyse de l'Ecole Polytechmque, Pans: Hermann. Volume 1 (1927); Volume 2 (1930). 7. The psychology of inventzon in the mathematzcal field, Princeton: Princeton Univ. Press (1945). (Traduction fran~aise. Paris, Albert Blanchard (1959)). 8. La th~ome des dquatzons aux d&ivdes partielles, P6kin, Editions Scientlfiques (1964). Universitd de Parzs-Sud Math~matzques-Bdtiment 425 91405 Orsay Cedex, France THE MATHEMATICAL INTELLIGENCER VOL 13, NO 1, 1991

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