Download e-book for kindle: Analyse complexe pour la Licence 3 : Cours et exercices by Patrice Tauvel

By Patrice Tauvel

ISBN-10: 2100500740

ISBN-13: 9782100500741

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4, si p ∈ N et z ∈ D(0, R) on a : S (p) (z) = ∞ (n + p)! an+p z n . n! ap . 1. Soit f une fonction définie dans un voisinage de z 0 ∈ C. On dit que f est développable en série entière au point z 0 s’il existe une série entière an z n , de rayon de convergence non nul, et un voisinage V de z 0 dans C tels que, pour tout z∈V : ∞ f (z) = n=0 an (z − z0 )n . 2. Soit f une fonction définie et indéfiniment dérivable dans un voisinage de 0. On appelle série de Mac-Laurin ou série de Taylor à l’origine de f la série entière 1 (n) f (0)z n .

Si |z| r , on a : r n z n z n |an z n | = an z0n M . 9. an z n converge pour z0 ∈ C, elle converge normalement dans tout disque D (0, r) tel que r < |z0 |. 2. Si la série entière Démonstration. Si la série donc est bornée. an z0n converge, la suite (an z0n )n admet 0 pour limite, an z n une série entière. Il existe un unique élément R de R+ ∪ {+∞} possédant les propriétés suivantes : (i) Pour tout z ∈ C tel que |z| < R, la série an z n est absolument convergente. (ii) Pour tout z ∈ C tel que |z| > R, la série an z n est divergente.

On a ainsi sup{|gn (t) − gp (t)| ; t ∈ I} sup{|fn (t) − fp (t)| ; t ∈ I}. Le résultat est donc une conséquence du critère de Cauchy uniforme. 5. Le réel x > 0 étant fixé, définissons gn : [0, 1] → R , t → (−1)n−1 f (t)enx(1−t) . n! Solutions des exercices 33 Si A = sup{|f (t)| ; t ∈ [0, 1]}, on a donc, pour 0 t A nx e =A · n! n! gn est normalement convergente sur [0, 1]. On en déduit |gn (t)| Ainsi, x étant fixé, la série en particulier que 1 0 1: (ex )n ∞ gn (t) dt = n=1 1 ∞ n=1 0 gn (t) dt, soit encore : 1 0 f (t) dt − 1 exp[−ex(1−t) ]f (t) dt = 0 1 ∞ (−1)n−1 n!

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Analyse complexe pour la Licence 3 : Cours et exercices corriges by Patrice Tauvel

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