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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2011 • Question 6

Question

Math 6210 • Real Analysis Spring 2011 • Question 6

Let \(f\) be a holomorphic function defined on a domain \(U\subset \mathbb C\). Let \[ \bar U=\{z\in \mathbb C\mid \bar z\in U\}. \] Consider the function \(g\) on \(\bar U\) defined by \[ g(z)=\overline {f(\bar z)}. \] Equivalently, \(g(z)\) is the complex conjugate of \(f(\bar z)\). Is \(g\) holomorphic or not? Explain your answer!

Raw LaTeX

Let \(f\) be a holomorphic function defined on a domain
\(U\subset\mathbb C\). Let
\[
\bar U=\{z\in\mathbb C\mid \bar z\in U\}.
\]
Consider the function \(g\) on \(\bar U\) defined by
\[
g(z)=\overline{f(\bar z)}.
\]
Equivalently, \(g(z)\) is the complex conjugate of \(f(\bar z)\). Is \(g\)
holomorphic or not? Explain your answer!

Topic Tags

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