Théorèmes de Picard

Pour l’article homonyme, voir Théorème du point fixe de Picard.

En analyse complexe, les théorèmes de Picard, du mathématicien Émile Picard, sont au nombre de deux :

Le petit théorème de Picard dit qu'une fonction entière non constante prend tout nombre complexe comme valeur, sauf peut-être un certain nombre complexe^[1].

Le grand théorème de Picard dit qu'une fonction holomorphe ayant une singularité essentielle prend, sur tout voisinage de cette singularité, tout nombre complexe une infinité de fois comme valeur, sauf peut-être un certain nombre complexe.

Remarques

Le « sauf peut-être un » dans ces énoncés est nécessaire, comme le montrent les exemples suivants. La fonction entière $z\mapsto \mathrm {e} ^{z}$ (l'exponentielle complexe) ne s'annule pas. Elle possède même une singularité essentielle en l'infini (c'est une fonction transcendante). La fonction $z\mapsto \mathrm {e} ^{1/z}$ est un exemple de fonction ne s'annulant pas avec singularité essentielle bornée (au point $0$ ).
Le cas des fonctions polynomiales est une conséquence directe du théorème de d'Alembert-Gauss.
Le petit théorème se déduit immédiatement du grand, car toute fonction entière est soit polynôme soit elle possède une singularité essentielle à l'infini.
Le grand théorème de Picard généralise le théorème de Weierstrass-Casorati.

Une récente conjecture de B. Elsner^[2] est liée au grand théorème de Picard : soient $U_{1},U_{2},\dots ,U_{n}$ des ouverts connexes qui ont pour réunion le disque unité épointé $D\setminus \{0\}$ . Sur chaque ouvert $U_{j}$ , soit $f_{j}$ une fonction holomorphe injective telle que $\mathrm {d} f_{j}=\mathrm {d} f_{k}$ sur toutes les intersections $U_{j}\cap U_{k}$ . Alors ces différentielles se recollent en une 1-forme méromorphe sur le disque $D$ . (Si le résidu est nul, la conjecture découle du grand théorème de Picard.)

Note

↑ (en) Serge Lang, Complex Analysis, vol. 103, Springer Verlag, coll. « Graduate Texts in Mathematics », 1985, 2^e éd., 367 p. (ISBN 978-1-4757-1871-3, lire en ligne), p. 343
↑ P. 330 de (en) Bernhard Elsner, « Hyperelliptic action integral », Annales de l'Institut Fourier, vol. 49, n^o 1,‎ 1999, p. 303-331 (lire en ligne).

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