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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2009 • Question 8

Question

Math 6210 • Real Analysis Fall 2009 • Question 8

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

(a) \(f\) is continuous;

(b) \(f\) is a 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 \(\mathbb C\).

Raw LaTeX

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

(a) \(f\) is continuous;

(b) \(f\) is a 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
\(\mathbb C\).

Topic Tags

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