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4. Prove the converse to Runge's theorem: if K is a compact set whose complement if not connected, then there exists a f
Posted: Thu May 12, 2022 1:31 pm
by answerhappygod
- 4 Prove The Converse To Runge S Theorem If K Is A Compact Set Whose Complement If Not Connected Then There Exists A F 1 (19.32 KiB) Viewed 43 times
4. Prove the converse to Runge's theorem: if K is a compact set whose complement if not connected, then there exists a function / holomorphic in a neighborhood of K which cannot be approximated uniformly by polynomial on K. [Hint: Pick a point zo in a bounded component of K, and let f(x) = 1/(z - zo). If f can be approximated uniformly by polynomials on K, show that there exists a polynomial p such that |(z - zo)p(2) - 11 <1. Use the maximum modulus principle (Chapter 3) to show that this inequality continues to hold for all z in the component of Kº that contains zo.]