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Using the branch Logo of logarithm, we define branches of two complex powers: a branch f(2)=z¹/2, and a branch g(z) of z¹/4, which are holomorphic on Do=C\{z EC: z=r, r>0}. i. What is f(Do), the image under f of Do? ii. Is it true that (g(z))² = f(z) for all z € Do? iii. Is it true that g(2²) = f(z) for all z € Do?

Using the branch Logo of logarithm, we define branches of two complex powers: a branch f(2)=z¹/2, and a branch g(z) of z¹/4, which are holomorphic on Do=C\{z EC: z=r, r>0}. i. What is f(Do), the image under f of Do? ii. Is it true that (g(z))² = f(z) for all z € Do? iii. Is it true that g(2²) = f(z) for all z € Do?

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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Using the branch Logo of logarithm, we define branches of two complex powers: a branch f(z)=z¹/2, and a branch g(z) of z¹/4, which are holomorphic on Do = C\ {z EC: z=r, r>0}. i. What is f(Do), the image under f of Do? ii. Is it true that (g(z))2 = f(z) for all z € Do? iii. Is it true that g(2²) = f(z) for all z € Do?
Transcribed Image Text:Using the branch Logo of logarithm, we define branches of two complex powers: a branch f(z)=z¹/2, and a branch g(z) of z¹/4, which are holomorphic on Do = C\ {z EC: z=r, r>0}. i. What is f(Do), the image under f of Do? ii. Is it true that (g(z))2 = f(z) for all z € Do? iii. Is it true that g(2²) = f(z) for all z € Do?
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