Let p: U → C be a holomorphic function on the star-shaped open set U. Suppose also p′(z) is continuous. Show that there exists a holomorphic function t: U → C such that p(z) = exp(t(z)) for all z ∈ U. Also show that t(z) is unique up to the addition of a constant 2πik, k ∈ Z. Use the complex FTC to prove this existence

Let p: U → C be a holomorphic function on the star-shaped open set U. Suppose also p′(z) is continuous. Show that there exists a holomorphic function t: U → C such that p(z) = exp(t(z)) for all z ∈ U. Also show that t(z) is unique up to the addition of a constant 2πik, k ∈ Z. Use the complex FTC to prove this existence

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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