Question 10.10. Prove that non-vanishing functions on simply connected domains have holomorphic square roots.
Question 10.10. Prove that non-vanishing functions on simply connected domains have holomorphic square roots.
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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Transcribed Image Text:Question 10.10. Prove that non-vanishing functions on simply connected domains have
holomorphic square roots.
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Step 1: Conceptual introduction
VIEWStep 2: Assume f(z) is Non-Vanishing and Holomorphic in D
VIEWStep 3: Represent f(z) in Polar Form
VIEWStep 4: Create a Continuous Argument Function
VIEWStep 5: Define the Square Root Function g(z)
VIEWStep 6: Prove g(z) is Holomorphic
VIEWStep 7: Complete the proof
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