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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2016 • Question 9

Question

Math 6210 • Real Analysis Spring 2016 • Question 9

Let \(f:U\to \mathbb C\) be a holomorphic function, where \(U\) is an open set. Let \(a\in U\). A standard result in complex analysis is that if \(f'(a)\ne 0\), then there is an open neighborhood \(V\subset U\) of \(a\) on which \(f\) is injective and the image \(f(V)\) is a neighborhood of \(f(a)\). Prove this result!

Raw LaTeX

Let \(f:U\to\mathbb C\) be a holomorphic function, where \(U\) is an
open set. Let \(a\in U\). A standard result in complex analysis is that if
\(f'(a)\ne0\), then there is an open neighborhood \(V\subset U\) of \(a\) on which
\(f\) is injective and the image \(f(V)\) is a neighborhood of \(f(a)\). Prove this
result!

Topic Tags

Automorphism Groups and Biholomorphisms Fourier Analysis and Convolution Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms