MATH502_HW7

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

• Notes
• 3

This preview shows pages 1–2. Sign up to view the full content.

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 ( *

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.
• Summer '08
• JOHNRIDENER
• Math, Topology, Sets, Continuous function, Metric space, Topological space, biholomorphic equivalence

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern