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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcee8942c-e69a-487b-a05a-79aff5c61f9f%2Fd21169fa-39e3-478f-a404-ea70362562d6%2Fvqo6968_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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