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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2019 • Question 6

Question

Math 6210 • Real Analysis Fall 2019 • Question 6

Prove the following special case of Runge’s approximation theorem: If \(f:B(0,4)\setminus (B(0,.1)\cup B(1,.1))\) is holomorphic and \(\epsilon >0\) is given, then there exists a rational function \(q\) so that \(|f(z)-q(z)|<\epsilon \) for all \[ z\in \overline {B(0,2)\setminus (B(0,.2)\cup B(1,.2))}. \]

Raw LaTeX

Prove the following special case of Runge's approximation
theorem: If \(f:B(0,4)\setminus(B(0,.1)\cup B(1,.1))\) is holomorphic and
\(\epsilon>0\) is given, then there exists a rational function \(q\) so that
\(|f(z)-q(z)|<\epsilon\) for all
\[
z\in\overline{B(0,2)\setminus(B(0,.2)\cup B(1,.2))}.
\]

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Meromorphic Functions and Rational Functions Runge Approximation