- 4 Do Not Use Liouville S Theorem For Any Parts Of This Question 4a 4 Points Let D C C Be A Domain And Zo D Supp 1 (62.91 KiB) Viewed 25 times
4. Do not use Liouville's theorem for any parts of this question. (4a) (4 points) Let D C C be a domain and zo € D. Suppose f: D\{zo} →→C is analytic and has a removable (z-%o)f(z), z‡ %0 is %= %0 discontinuity at zo. Prove from the definition of differentiability that g(z) = differentiable at zo. Conclude that it is holomorphic on D.. (4b) (3 points) Let & > 0. Suppose f: C\ {zo} → C is analytic and that there exists M > 0 such that f(z) dz=0 for all w # %0. |z-zo|>M implies that f(z)| <- Prove that lim |z-zol=r 2-w (4c) (3 points) Let &> 0 and let zo E C. Suppose f: C\ {zo} → C is analytic and has a removable discontinuity at zo. Use parts (a) and (b) to prove that if there exists MER such that f(z) <₁¹ for all z EC with |z-zol > M, then f(z) = 0 for all z = C\ {z}. (Hint: compute the integral in part (b) in another way for r > |zo-w.)