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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Fall 2022 • Question 3

Question

Math 6220 • Complex Analysis Fall 2022 • Question 3

Let \(\Omega \) be a bounded domain in \(\mathbb {C}\) and \(f_n:\Omega \to \mathbb {C}\) a sequence of holomorphic functions such that \[ \int _\Omega |f_n-f_m|^2\,dA(z)\to 0\text { as }n,m\to \infty . \] Here \(dA(z)\) is the Lebesgue area integral. Show that there is \(f\) holomorphic on \(\Omega \) such that \(f_n\to f\) locally uniformly on \(\Omega \).

Raw LaTeX

Let \(\Omega\) be a bounded domain in \(\mathbb{C}\) and \(f_n:\Omega\to\mathbb{C}\) a sequence of holomorphic
functions such that
\[
\int_\Omega |f_n-f_m|^2\,dA(z)\to 0\text{ as }n,m\to\infty.
\]
Here \(dA(z)\) is the Lebesgue area integral. Show that there is \(f\) holomorphic
on \(\Omega\) such that \(f_n\to f\) locally uniformly on \(\Omega\).

Topic Tags

Cauchy Integral Theory and Morera's Theorem Complex Integration and Real Integral Evaluation Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms