Émile Picard

Émile Picard (1856–1941) was a French mathematician whose work shaped complex analysis, differential equations, and algebraic geometry. Picard’s theorems describe how entire and meromorphic functions can omit values.

Complex analysis: The study of functions of complex variables and their analytic properties. Picard’s most famous theorems concern how holomorphic functions behave.
Little Picard theorem: A theorem stating that a nonconstant entire function omits at most one complex value. It is Picard’s striking global restriction on entire functions.
Charles Hermite: A French mathematician known for work in algebra, number theory, and analysis. Hermite was Picard’s doctoral adviser and an important influence on his mathematical formation.
Picard–Vessiot theory: A differential Galois theory describing algebraic relations among solutions of linear differential equations. Named in part for Picard, it connects differential equations with algebraic groups.
Liouville's theorem: A theorem stating that every bounded entire function is constant. It restricts entire functions by boundedness, whereas Picard restricts them by omitted values.
Holomorphic function: A complex function that is complex differentiable throughout an open domain. Picard’s value-omission results apply to holomorphic functions and their extensions.
Great Picard theorem: A theorem stating that near an essential singularity, a meromorphic function takes every value with at most two exceptions infinitely often. It gives the local counterpart to the little theorem’s value restriction.
Henri Poincaré: A French mathematician whose work helped establish topology, dynamical systems, and modern mathematical physics. Picard and Poincaré were leading contemporaries in French mathematics.
Picard group: A group of line bundles or divisor classes associated with an algebraic variety. Picard’s work helped establish the study of divisor classes that this invariant organizes.
Casorati–Weierstrass theorem: A theorem stating that values near an essential singularity are dense in the complex plane. Picard’s theorem strengthens this density conclusion to nearly all values, attained infinitely often.

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