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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2009 • Question 8

Question

Math 6210 • Real Analysis Spring 2009 • Question 8

Let \(f:\Omega \to \mathbb C\) be a complex-valued function on an open domain \(\Omega \subset \mathbb C\). Assume that for all closed contours \(\gamma \) that consists of line segments parallel to the real or imaginary axes \[ \int _\gamma f(z)\,dz=0. \] Show that there exists a holomorphic function \(F:\Omega \to \mathbb C\) such that \(F'(z)=f(z)\).

Raw LaTeX

Let \(f:\Omega\to\mathbb C\) be a complex-valued function on an open
domain \(\Omega\subset\mathbb C\). Assume that for all closed contours \(\gamma\)
that consists of line segments parallel to the real or imaginary axes
\[
\int_\gamma f(z)\,dz=0.
\]
Show that there exists a holomorphic function \(F:\Omega\to\mathbb C\) such that
\(F'(z)=f(z)\).

Topic Tags

Complex Integration and Real Integral Evaluation Holomorphic Functions and Cauchy-Riemann Equations Lebesgue Integration Open Mapping and Closed Graph Theorems