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