crimboy[18] 复分析...

题目背景

证明连续性时缩放水平哈哈了(sweatgrinning)

题面

f is a nonconstant holomorphic function on a region \Omega_1 and g is defined on f(\Omega_1). Prove that if h=g\circ f is holomorphic, then so is g.

先想再看提示哦

提示

f is nonconstant so Z(f') has only isolated points. At y_0=f(x_0) where x_0\notin Z(f') g obviously has a derivative. At y_0=f(x_0) where x_0\in Z(f') we try to prove g is continuous. Then the integration of g over any triangle in a neighborhood U of x_0 where g is holomorphic on U-\lbrace x_0\rbrace is 0. Then g is also holomorphic on U by Morera's theorem. How to prove continuity then? Please use the result that f(x)=f(x_0)+(\varphi(x))^m for some open one-to-one \varphi in some neighborhood of x_0.

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