French mathematician whose work was mainly in the fields of mathematical analysis and algebraic geometry. He formulated two theorems on integral functions. He applied mathematical principles as much as possible to other branches of science, particularly physics and engineering.
Picard was born in Paris and studied at the Ecole Normale Supérieure. At the age of 23 he was appointed professor in Toulouse, but returned to Paris two years later and became professor at the Sorbonne in 1885.
Picard's ‘little theorem’ states that an integral function of the complex variable takes every finite value, with one possible exception. In 1879 he expressed it in this way:
Let f(z) be an entire function. If there exist two values of A for which the equation f(z) = A does not have a finite root, then f(z) is a constant. From this it follows that if f(z) is an entire function that is not a constant, there cannot be more than one value of A for which f(z) = A has no solution.
This was followed 1880 by Picard's ‘big theorem’:
Let f(z) be a function, analytic everywhere except at a where it has an essential isolated singularity; the equation f(z) = A has in general an infinity of roots in any neighbourhood of a. Although the equation can fail for certain exceptional values of the constant A, there cannot be more than two such values.
Picard's work on the integrals attached to algebraic surfaces, together with the associated topological questions, developed into an area of algebraic geometry that had applications in topology and function theory.
Much of Picard's work was recorded in the three-volume Traité d'analyse.