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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Summer 2021 • Question 7

Question

Math 6220 • Complex Analysis Summer 2021 • Question 7

Suppose that \(f\) is holomorphic on the upper half plane \(\mathbb {H}=\{\operatorname {Im}(z)>0\}\), \(f\) extends continuously to \(\overline {\mathbb {H}}=\mathbb {H}\cup \mathbb {R}\), and \(f(\mathbb {R})\subset \mathbb {R}\). Show that \(f\) has a unique holomorphic extension to \(\mathbb {C}\) and explicitly identify that extension (in terms of \(f\)).

Raw LaTeX

Suppose that \(f\) is holomorphic on the upper half plane \(\mathbb{H}=\{\operatorname{Im}(z)>0\}\),
\(f\) extends continuously to \(\overline{\mathbb{H}}=\mathbb{H}\cup\mathbb{R}\), and \(f(\mathbb{R})\subset\mathbb{R}\). Show that \(f\) has a
unique holomorphic extension to \(\mathbb{C}\) and explicitly identify that extension
(in terms of \(f\)).

Topic Tags

Argument Principle and Rouche's Theorem Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Maximum Modulus Principle