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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2006 • Question 6

Question

Math 6210 • Real Analysis 2006 • Question 6

Let \(f\) be a holomorphic function on an open set containing \(\overline {D_1}(0)\). Prove that if \(z\) is in \(D_1(0)\), then \[ f(z)=\frac {1}{2\pi i}\int _\gamma \frac {f(w)}{\sin (w-z)}\,dw \] where \(z\in D_1(0)\) and \(\gamma \) traverses the unit circle once in the positive direction.

Raw LaTeX

Let \(f\) be a holomorphic function on an open set containing
\(\overline{D_1}(0)\). Prove that if \(z\) is in \(D_1(0)\), then
\[
f(z)=\frac{1}{2\pi i}\int_\gamma\frac{f(w)}{\sin(w-z)}\,dw
\]
where \(z\in D_1(0)\) and \(\gamma\) traverses the unit circle once in the positive
direction.

Topic Tags

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