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4. (Zeros and singularities) (a) Determine all funetion f analytic in |2| <1 and satisfying (4) -- n= 2,3, 4, . (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(:)| Se" for |2| 2 1, z =z+iy. Determine f. [Hint: Use the version of Liouville's theorem from Problem 8 (a) in Homework assignment 1.)

4. (Zeros and singularities) (a) Determine all funetion f analytic in |2| <1 and satisfying (4) -- n= 2,3, 4, . (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(:)| Se" for |2| 2 1, z =z+iy. Determine f. [Hint: Use the version of Liouville's theorem from Problem 8 (a) in Homework assignment 1.)

Advanced Engineering Mathematics
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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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.)
Transcribed Image Text: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.)
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.
Transcribed Image Text: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.
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