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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2004 • Question 10

Question

Math 6210 • Real Analysis 2004 • Question 10

Let \(f\) be a holomorphic automorphism of \(\mathbb C^*\). Show that either \(f(z)=\alpha z\) or \(f(z)=\frac {\alpha }{z}\) for some complex number \(\alpha \ne 0\).

Raw LaTeX

Let \(f\) be a holomorphic automorphism of \(\mathbb C^*\). Show that
either \(f(z)=\alpha z\) or \(f(z)=\frac{\alpha}{z}\) for some complex number
\(\alpha\ne 0\).

Topic Tags

Automorphism Groups and Biholomorphisms Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Zeros of Holomorphic Functions and the Identity Theorem