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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Fall 2021 • Question 5

Question

Math 6220 • Complex Analysis Fall 2021 • Question 5

Prove the following special case of Runge’s approximation theorem: If

\[ f:B(0,4)\setminus (B(0,.1)\cup B(1,.1))\to \mathbb {C} \]

is holomorphic and \(\epsilon >0\) is given then there exists \(q\), a rational function, so that

\[ |f(z)-q(z)|<\epsilon \text { for all }z\in B(0,2)\setminus (B(0,.2)\cup B(1,.2)). \]

Raw LaTeX

Prove the following special case of Runge's approximation theorem: If

\[
f:B(0,4)\setminus(B(0,.1)\cup B(1,.1))\to\mathbb{C}
\]

is holomorphic and \(\epsilon>0\) is given then there exists \(q\), a rational function, so that

\[
|f(z)-q(z)|<\epsilon\text{ for all }z\in B(0,2)\setminus(B(0,.2)\cup B(1,.2)).
\]

Topic Tags

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