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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Summer 2023 • Question 5

Question

Math 6220 • Complex Analysis Summer 2023 • Question 5

Let \(f\) be holomorphic on \(\mathbb {C}\setminus \{0,2\}\) with simple poles at 0 and 2. Suppose that \(\int _{|z|=1} f(z)\,dz=2\pi i\) and \(\int _{|z|=3} f(z)\,dz=0\). Also suppose that \(f\) is bounded on \(|z| > 3\). List all such functions (with a proof that the list is complete).

Raw LaTeX

Let \(f\) be holomorphic on \(\mathbb{C}\setminus\{0,2\}\) with simple poles at 0 and 2. Suppose
that \(\int_{|z|=1} f(z)\,dz=2\pi i\) and \(\int_{|z|=3} f(z)\,dz=0\). Also suppose that \(f\) is
bounded on \(|z| > 3\). List all such functions (with a proof that the list is
complete).

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Isolated Singularities and Residues Zeros of Holomorphic Functions and the Identity Theorem