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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2004 • Question 9

Question

Math 6210 • Real Analysis 2004 • Question 9

Let \(\Omega \) open in \(\mathbb C\), \(b\in \Omega \), \(f\in H(\Omega )\), and assume that \(f'(b)\ne 0\). Show that \[ \frac {2\pi i}{f'(b)}=\int _C \frac {1}{f(z)-f(b)}\,dz \] for sufficiently small positively oriented simple circles centered at \(b\).

Raw LaTeX

Let \(\Omega\) open in \(\mathbb C\), \(b\in \Omega\), \(f\in H(\Omega)\),
and assume that \(f'(b)\ne 0\). Show that
\[
\frac{2\pi i}{f'(b)}=\int_C \frac{1}{f(z)-f(b)}\,dz
\]
for sufficiently small positively oriented simple circles centered at \(b\).

Topic Tags

Lebesgue Integration