VI.65 Jacques Hadamard

b. Versailles, France, 1865; d. Paris, 1963
Function theory; calculus of variations; number theory;
partial differential equations; hydrodynamics

A graduate of the École Normale in Paris, Hadamard obtained a position at the University of Bordeaux in 1893. He returned to Paris in 1897 where he taught at the Collège de France, the École Polytechnique, and the École Centrale until his retirement in 1937. The Hadamard Seminar at the Collège de France, where mathematicians came from around the world to expound on recent results, was an influential and integral part of mathematical life in France between the wars.

Hadamard’s first significant papers were concerned with the theory of HOLOMORPHIC FUNCTIONS [I.3 §5.6] of a complex variable, in particular with the analytic continuation of a Taylor series; and in his thesis of 1892 he investigated how the properties of the singularities of a series could be deduced from those of its coefficients. Notably he showed that the radius of convergence R of a Taylor series could be given by , a result now known as the Cauchy-Hadamard theorem. (CAUCHY [VI.29] had published the formula in 1821 but Hadamard, who had discovered it independently, was the first to give a complete proof.) Further results followed, including the famous “Hadamard gap theorem,” which gives the condition for the circle of convergence of the series to be a natural boundary of the function. His monograph La Série de Taylor et son Prolongement Analytique (1901) proved especially influential. In 1912 he formulated the problem of quasi-analyticity for infinitely differentiable functions.

The year 1892 also saw the appearance of Hadamard’s prize-winning memoir on entire functions, in which he used results from his thesis to establish the relations between the coefficients of the Taylor series of an entire function and its zeros, and then applied them to evaluate the genus of the entire function. He applied this work, and other results from his thesis, to the RIEMANN ZETA FUNCTION [IV.2 §3], which enabled him, in 1896, to prove his most famous result: the PRIME NUMBER THEOREM [V.26]. (The theorem was proved simultaneously by DE LA VALLÉE POUSSIN [VI.67] but in a more complicated way.)

Hadamard’s other key achievements of the 1890s include a well-known inequality concerning DETERMINANTS [III.15] (1893), a result essential in the FRED-HOLM THEORY [IV.15 §1] of integral equations; and his “three-circles theorem” (1896), which demonstrates the importance of convexity in the study of analytic functions and plays a significant role in interpolation theory.

In 1896 Hadamard won the Prix Bordin for his study of the behavior of geodesics on surfaces. (The motivation for studying such geodesics is that they can be used to represent the trajectories of motion in dynamical systems.) It was Hadamard’s first major work on a subject other than analysis. His two papers, one on geodesics on a surface of positive curvature (1897) and the other on geodesics on a surface of negative curvature (1898), are characterized by a qualitative analysis inherited from POINCARÉ [VI.61]. The first relies on results from classical differential geometry, while the second is dominated by topological considerations.

Prompted by an interest in the CALCULUS OF VARIATIONS [III.94], Hadamard developed the ideas of Volterra’s functional calculus. In 1903 he was the first to describe linear functionals on a function space. By considering the space of continuous functions on a given interval, he showed that every functional is the limit of a sequence of intervals, a result now recognized as a precursor to the RIESZ REPRESENTATION THEOREM [III.18] formulated by RIESZ [VI.74] in 1909. Hadamard’s influential Leçons sur le Calcul de Variations (1910) is the first book in which the ideas of modern functional analysis can be found.

In applied mathematics Hadamard was primarily concerned with wave propagation, in particular highspeed flows. In 1900 he began working on the theory of partial differential equations, and in 1903 published Leçons sur la Propagation des Ondes et les Équations de l’Hydrodynamique; this was followed by Lectures on Cauchy’s Problem in Linear Partial Differential Equations (1922). The latter contained the details of his fundamental idea of the WELL-POSED PROBLEM [IV.12 §2.4] (i.e., a problem in which the solution must not only exist and be unique but must also depend continuously on the initial data). The origins of the idea can be found in his 1898 paper on GEODESICS [I.3 §6.10].

Hadamard’s book The Psychology of Invention in the Mathematical Field (1945) is well-known for its discussion of the unconscious and its role in mathematical discovery.

Further Reading

Hadamard, J. 1968. Collected Works: Œuvres de Jacques Hadamard, four volumes. Paris: CNRS.

Maz’ya, V., and T. Shaposhnikova. 1998. Jacques Hadamard. A Universal Mathematician. Providence, RI: American Mathematical Society/London Mathematical Society.