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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Fall 2022 • Question 5

Question

Math 6220 • Complex Analysis Fall 2022 • Question 5

Prove the following special case of Runge’s theorem: Suppose that \(f:D(0,4)\setminus (\overline {D(1,.1)}\cup \overline {D(-1,.1)})\to \mathbb {C}\) is holomorphic and \(\epsilon >0\) is given. Then there is a rational function \(q\) such that \(|f(z)-q(z)|\le \epsilon \) for \(z\in K=\) \[ \overline {D(0,2)}\setminus (\overline {D(1,.2)}\cup \overline {D(-1,.2)}). \]

Raw LaTeX

Prove the following special case of Runge's theorem: Suppose that
\(f:D(0,4)\setminus(\overline{D(1,.1)}\cup\overline{D(-1,.1)})\to\mathbb{C}\) is holomorphic and \(\epsilon>0\) is given.
Then there is a rational function \(q\) such that \(|f(z)-q(z)|\le\epsilon\) for \(z\in K=\)
\[
\overline{D(0,2)}\setminus(\overline{D(1,.2)}\cup\overline{D(-1,.2)}).
\]

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Meromorphic Functions and Rational Functions Zeros of Holomorphic Functions and the Identity Theorem