Answered: 8. (Liouville's theorem, Cauchy… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section: Chapter Questions

Problem 8RE

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8. (Liouville's theorem, Cauchy estimates and the maximum modulus principle)
(a) Suppose that f is entire, and that for some positive integer k and some constants C,r > 0 it
holds that
\S(2)| < C\z|*,
|=| 2 r.
Show that f is a polynomial of degree at most k.

4. (Zeros and singularities)
(a) Determine all function f analytic in |2| < 1 and satisfying
n= 2,3, 4, .
n+1
(b) The function f is analytic in the whole complex plane with the exception of a simple pole at
the origin with residue i. Moreover, f(i) = e and |f(z)| < e® for |2| 2 1, z = 1+ iy. Determine f.
[Hint: Use the version of Liouville's theorem from Problem 8 (a) in Homework assignment 1.)

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