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4). 2. En utilisant la règle de Leibniz , on trouve 9 \Pv ( l 9f -i-1 f\2dp)] - 9p v ( l 9f , I f\2dp\ d fd f +l 7f . d2f _ m2 d2^ 7 d ld p df dp '■dz dz J dzdz +' I[J/ I12 dzdz '' JJ dfzdz ' JJ dzdz + ep \ iL iL + / ^ U ^ Z + 7 ^ j + e P|f |2 Ü L = 'dz d z ''d z dz' dzdz + é > ( i l L , 7 ° e °L , 19 19 \ _ # * 1 * 1 . 18) Formes différentielles dans le plan 20 On utilise ensuite le lemme de Schwarz pour affirmer que d2f dzdz d2f dzdz puis les identités W = d£ dz dz (voir exercice 1 . 1 ) et dp _ dp dz dz (p étant une fonction réelle).

3 Soit £ n>o an X n une série entière de rayon de convergence R > 0. La fonction f définie dans le disque ouvert {z G € ; \z — z0\ < R} par f ( z ) '■= Y , °»(* ~ zo)n n>0 est holomorphe dans ce disque. Preuve, Considérons 0 < r < R ; dans le disque fermé D (z0, r), la fonction / est limite uniforme de la suite de polynômes Pn{z) = Y ^ ak(z ~ zo)k fc=0 en la variable complexe z. 2, pour tout n G N, pour tout z dans le disque ouvert {z G C ; \z — z0\ < r}, P n( C) Ç -z d f. Fixons maintenant 2 dans {z G C ; \z — zo\ < r}.

Formes différentielles dans le plan 16 F ig . 3) et le fait que les aller-retours le long d’un même 1 -simplexe se détruisent dans le calcul de l’intégrale curviligne, on voit que — ( [ Ü Q -d Ç - [ ■ ^Q -dA = — [[ dÇAdÇ. z Ç , - Z ) JJD(zo, R)\D (z,t) df 1 dÇ Ç - Z dÇAdÇ. 13). 9 *Voici un exercice que nous retrouverons dans la suite du cours. Soit

*(z) := “™ J/J[R 2 v{$+irl)Çc+^ IT) — Z.

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