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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2009 • Question 6

Question

Math 6210 • Real Analysis Spring 2009 • Question 6

Let \(f\) be a holomorphic function on \(\mathbb C\setminus \{0\}\) such that \(f(z)\sin z\to 1\) as \(z\to 0\). Calculate \[ \int _\gamma f(z)\,dz \] where \(\gamma \) is the unit circle traversed counter-clockwise.

Raw LaTeX

Let \(f\) be a holomorphic function on \(\mathbb C\setminus\{0\}\)
such that \(f(z)\sin z\to1\) as \(z\to0\). Calculate
\[
\int_\gamma f(z)\,dz
\]
where \(\gamma\) is the unit circle traversed counter-clockwise.

Topic Tags

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