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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods

Fall 2023 • Question 2

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods Fall 2023 • Question 2

Prove the argument principle: Assume that \(f\) is a meromorphic function inside and on a simple closed contour \(C\) (oriented in the typical counter-clockwise fashion), that \(f\) has exactly \(N\) zeros (counting multiplicity) in this region all lying within \(C\), and that \(f\) has exactly \(M\) poles (counting order), all lying within \(C\). Prove that \[ \frac {1}{2\pi i}\oint _C \frac {f'(z)}{f(z)}\,dz=N-M. \]

Raw LaTeX

Prove the argument principle: Assume that \(f\) is a meromorphic function inside and on
a simple closed contour \(C\) (oriented in the typical counter-clockwise fashion), that \(f\) has
exactly \(N\) zeros (counting multiplicity) in this region all lying within \(C\), and that \(f\) has
exactly \(M\) poles (counting order), all lying within \(C\). Prove that
\[
\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}\,dz=N-M.
\]

Topic Tags

Cauchy Integral Formula and Integral Theorems Contour Integration and Residue Calculus Isolated Singularities and Residues Meromorphic Functions and the Argument Principle