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Let f(z) be holomorphic on C (entire function). (a) Show that for any a, b e C, there exists R> 0 such that f(a) f(b): = a-b 2πi f(z) $₁ (z − a) (² — b) dz, where y is C(0, R) oriented positively. (b) If f(z) is bounded, prove that (actually) f(z) is constant.

Let f(z) be holomorphic on C (entire function). (a) Show that for any a, b e C, there exists R> 0 such that f(a) f(b): = a-b 2πi f(z) $₁ (z − a) (² — b) dz, where y is C(0, R) oriented positively. (b) If f(z) is bounded, prove that (actually) f(z) is constant.

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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please answer letter b. instead of using Cauchy inequality, use Louville’s theorem
Let f(z) be holomorphic on C (entire function). (a) Show that for any a, b e C, there exists R> 0 such that f(a) f(b) = a-b f(z) $.00 2πί (z-a)(z-b) dz, where y is C(0, R) oriented positively. (b) If f(z) is bounded, prove that (actually) f(z) is constant.
Transcribed Image Text:Let f(z) be holomorphic on C (entire function). (a) Show that for any a, b e C, there exists R> 0 such that f(a) f(b) = a-b f(z) $.00 2πί (z-a)(z-b) dz, where y is C(0, R) oriented positively. (b) If f(z) is bounded, prove that (actually) f(z) is constant.
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