Suppose that f is holomorphic on the whole of C and suppose that |f(z)| < K|z|k for some real constant K > 0 and some positive integer k > 0. Prove that f is a polynomial function of degree at most k.

Suppose that f is holomorphic on the whole of C and suppose that |f(z)| < K|z|k for some real constant K > 0 and some positive integer k > 0. Prove that f is a polynomial function of degree at most k.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.3: Change Of Basis

Problem 17EQ

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Suppose that f is holomorphic on the whole of C and suppose that |f(z)| < K|z|k
for some real constant K > 0 and some positive integer k > 0. Prove that f is a
polynomial function of degree at most k.
[Hint: Calculate the coefficients of z", n > k = 1, in the Taylor expansion of f
around 0].
6/9

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