MATH502_HW7 - Serge Ballif MATH 502 Homework 7(1(The...

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Serge Ballif MATH 502 Homework 7 March 21, 2008 (1) (The inverse function theorem) Let U and V be open subsets of and let f : U V be a holomorphic bijection. Show that f is a homeomorphism (that is, show that the inverse function f - 1 is continuous; use the open mapping theorem). Show further that f is a biholomorphic equivalence (that is, show that f - 1 is differentiable). Since f is a bijection, the inverse function f - 1 : V U exists and is also a bijection. For simplicity of notation, rename f - 1 = g . Claim 1. g is continuous. Proof. Since f is a bijection, each subset of U can be written in the form g ( W ) for some unique subset W V . Note that f is a nonconstant holomorphic map on an open set. Thus, f is an open map (by the open mapping theorem). Therefore, if g ( W ) is an open subset of U , g - 1 ( g ( W )) = f ( g ( W )) = W is an open subset of V . Therefore, g is a continuous function. Claim 2. g is differentiable. Proof. At each w U , f has some power series representation f ( z ) = f ( w ) + c 1 ( z - w ) + c 2 ( z - w ) 2 + · · · because f is holomorphic on U . Some algebraic manipulation yields the equation f ( z ) - f ( w ) z - w = h ( z ) ( * ) for some power series h ( z ). Taking the limit as z w of both sides of ( *
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