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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2017 • Question 9

Question

Math 6210 • Real Analysis Fall 2017 • Question 9

Let \(f:\mathbb C\to \mathbb C\) be a function such that:

(a) \(f\) is continuous;

(b) \(f\) is holomorphic at all points \(z\) not on the real axis;

(c) if \(z\) is on the real axis then \(f(z)\) is real.

Show that \(f(\bar z)=\overline {f(z)}\) and that \(f\) is holomorphic on all of \(\mathbb C\). In words, applying \(f\) to the conjugate input gives the complex conjugate of the original output.

Raw LaTeX

Let \(f:\mathbb C\to\mathbb C\) be a function such that:

(a) \(f\) is continuous;

(b) \(f\) is holomorphic at all points \(z\) not on the real axis;

(c) if \(z\) is on the real axis then \(f(z)\) is real.

Show that \(f(\bar z)=\overline{f(z)}\) and that \(f\) is holomorphic on all of
\(\mathbb C\). In words, applying \(f\) to the conjugate input gives the complex
conjugate of the original output.

Topic Tags

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