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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2008 • Question 8

Question

Math 6210 • Real Analysis Spring 2008 • Question 8

Let \(\Delta \) be the open unit disk and \(\gamma \) the unit circle in \(\mathbb C\) and \(\phi :\gamma \to \mathbb C\) a continuous function. Let \(g\) be a meromorphic function on \(\mathbb C\) that has a single simple pole at \(0\). Define a function \(f:\Delta \to \mathbb C\) by the formula \[ f(z)=\int _\gamma \phi (w)g(w-z)\,dw. \] Show that \(f\) is holomorphic.

Raw LaTeX

Let \(\Delta\) be the open unit disk and \(\gamma\) the unit circle in
\(\mathbb C\) and \(\phi:\gamma\to\mathbb C\) a continuous function. Let \(g\) be a
meromorphic function on \(\mathbb C\) that has a single simple pole at \(0\).
Define a function \(f:\Delta\to\mathbb C\) by the formula
\[
f(z)=\int_\gamma \phi(w)g(w-z)\,dw.
\]
Show that \(f\) is holomorphic.

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Isolated Singularities and Residues Lebesgue Integration Meromorphic Functions and Rational Functions