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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1994 • Question 10

Question

Math 6210 • Real Analysis Fall 1994 • Question 10

Let \(f\) be holomorphic in an open set containing the disk \(D=\{z\in \mathbb C:|z|\le r\}\) and suppose that the only zero of \(f\) in \(D\) is a simple zero at \(a\) belonging to the interior of \(D\). If \(C_r\) is the positively oriented contour \(|z|=r\) show that \[ \int _{C_r}\frac {1}{f(z)}\,dz=\frac {2\pi i}{f'(a)}. \]

Raw LaTeX

Let \(f\) be holomorphic in an open set containing the disk
\(D=\{z\in\mathbb C:|z|\le r\}\) and suppose that the only zero of \(f\) in \(D\) is a
simple zero at \(a\) belonging to the interior of \(D\). If \(C_r\) is the positively
oriented contour \(|z|=r\) show that
\[
\int_{C_r}\frac{1}{f(z)}\,dz=\frac{2\pi i}{f'(a)}.
\]

Topic Tags

Complex Integration and Real Integral Evaluation Holomorphic Functions and Cauchy-Riemann Equations Lebesgue Integration Zeros of Holomorphic Functions and the Identity Theorem