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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2014 • Question 7

Question

Math 6210 • Real Analysis Fall 2014 • Question 7

Let \(f\) be an a holomorphic function defined on an open set \(U\subset \mathbb C\) and \(a\in U\). Assume that \(f'(a)\) is nonzero. Show that for sufficiently small circles \(C\) centered at \(a\) \[ \frac {2\pi i}{f'(a)}=\int _C \frac {dz}{f(z)-f(a)} \]

Raw LaTeX

Let \(f\) be an a holomorphic function defined on an open set
\(U\subset\mathbb C\) and \(a\in U\). Assume that \(f'(a)\) is nonzero. Show that
for sufficiently small circles \(C\) centered at \(a\)
\[
\frac{2\pi i}{f'(a)}=\int_C \frac{dz}{f(z)-f(a)}
\]

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Lebesgue Integration Zeros of Holomorphic Functions and the Identity Theorem