Le grand théorème de Picard pour les multifonctions analytiques finies

Bernard Aupetit¹ ; Mustapha Ech-Chérif El Kettani²

¹Département de mathématiques et de statistique Faculté des sciences et de génie Université Laval Québec, Canada, G1K 7P4
²Département de mathématiques et informatique Faculté des sciences Dhar El-Mahraz B.P. 1796 Atlas Fès, Maroc

Annales Polonici Mathematici, Tome 77 (2001) no. 2, pp. 189-196

Résumé

Let $D$ be a domain of the complex plane containing the origin. The famous great theorem of Émile Picard asserts that if $h$ is holomorphic on $D\setminus\{ 0\} $, with an essential singularity at 0, then the image under $h$ of any pointed neighbourhood of 0 covers all the complex plane, with at most one exception. Introducing the concept of essential singularity for analytic multifunctions, we extend this theorem to a finite analytic multifunction $K$, of degree $N$, defined on $D\setminus\{ 0\}$. In this case $\bigcup _{0 |\lambda | r}K (\lambda )$ covers all the complex plane, with at most $2N-1$ exceptions. In particular, this theorem can be used in the case of $N\times N$ matrices whose entries are holomorphic on $D\setminus \{ 0\} $ with essential singularities at 0. In this case, if their spectra avoid $2N$ points on a pointed neighbourhood of 0, these spectra must be constant.

Détail

DOI : 10.4064/ap77-2-5

Mots-clés : domain complex plane containing origin famous great theorem mile picard asserts holomorphic setminus essential singularity image under pointed neighbourhood covers complex plane exception introducing concept essential singularity analytic multifunctions extend theorem finite analytic multifunction degree defined setminus bigcup lambda lambda covers complex plane n exceptions particular theorem times matrices whose entries holomorphic setminus essential singularities their spectra avoid points pointed neighbourhood these spectra constant