Picard was primarily an analyst. One of his earliest published papers (1879) contains the striking result from which the modern theory of integral functions of a complex variable has developed. Picard's theorem is that an integral function, i.e. one that has no singularity except at infinity, assumes every value with one exception at most. His proof depended on a property of the elliptic modular function, and many years elapsed before an "elementary" proof was discovered.
Much of Picard's work in the theory of functions is concerned with functions of two independent variables. In 1882 he established the existence of certain discontinuous groups of linear substitutions of two variables, and constructed functions invariant under the substitutions of these groups. Thus was created the theory of automorphic functions of two variables, which he elaborated in subsequent papers.
The name "Picard integrals," applied to integrals of total differentials associated with an algebraic surface, is an indication of Picard's share in the development of the theory of algebraic functions of two independent variables. In addition to many other papers, and the treatise mentioned later, his published work in this field includes a memoir - the first substantial contribution to the theory - which the Académic des Sciences recognised by the award of the Grand Prix des Sciences Mathématiques in 1888. His later work on this subject is closely interwoven with that of the geometers of the Italian school.
Picard made outstanding contributions to the theory of differential equations, especially the part that consists in establishing the existence of solutions. His main weapon here was the method of successive approximations due to H A Schwarz, which he applied with such skill, and to so many different kinds of problem, that it is commonly known as Picard's method. This process, which is applicable also to ordinary differential equations, was first used by Picard about 1890 in his study of partial differential equations of elliptic type (of which Laplace's equation is an important particular case). Many applications of the method are given in his very readable book on partial differential equations.
In further illustration of the range of Picard's researches may be mentioned his work on ordinary linear differential equations with doubly-periodic coefficients; his extension to the theory of ordinary linear differential equations (which is in some respects similar to that of algebraic equations) of the ideas of Galois; and his papers on algebraic forms, on integral equations, and on mathematical physics.
In addition to a very large number of papers published in various journals, Picard wrote several books: the Traité d'analyse, in three volumes; the Théorie des fonctions algébriques de deux variables indépendantes, in two volumes (in collaboration with G Simart); a number of shorter works on partial differential equations, functional equations, etc., based on courses of lectures, which take the place of a projected fourth volume of the Traité; and several volumes of miscellaneous papers, some of a philosophical nature.
Many honours were conferred on Picard, in his own country and abroad. He was elected to the Académie des Sciences at the age of thirty-three, and was for many years Secrétaire Perpétuel. He was twice President of the Société mathématique de France, a rare distinction; and in 1923 he was President of the Société française de Physique. Shortly before reaching the age of seventy he was elected to the Académie Française. In this country he was a Foreign Member of the Royal Society, and had been an Honorary Fellow of this Society since 1920.
On the occasion of the celebration of his scientific jubilee in 1928 the address of the Society, which was represented at the Sorbonne by Professor E T Whittaker, referred to its distinguished Honorary Fellow in the following terms: "Chercheur éminent, esprit capable de traiter avec autorité des idées les plus abstraites, écrivain délicieux, il personnifie ce que l'étranger a appris à associer de mieux au nom d'un homme de science français."
Picard died on December 12, 1941.