Let V = C {z < 0}. Define the principal logarithm Log: V → C as he branch of log on V such that Log(1) = 0. (a) Prove that such a branch of log exists on V. (b) Prove for z E V that Log(z) = ln |z| + iArg(2), where In is the natural logarithm on positive reals, and Arg is the principal argument with image (-n, 7]. (c) Is Log(zw) = Log(z) + Log(w) for all z, w e V? Justify your answer. Similarly, is Log(z") = nLog(z) for all n E Z and z E V? Justify your answer.
Let V = C {z < 0}. Define the principal logarithm Log: V → C as he branch of log on V such that Log(1) = 0. (a) Prove that such a branch of log exists on V. (b) Prove for z E V that Log(z) = ln |z| + iArg(2), where In is the natural logarithm on positive reals, and Arg is the principal argument with image (-n, 7]. (c) Is Log(zw) = Log(z) + Log(w) for all z, w e V? Justify your answer. Similarly, is Log(z") = nLog(z) for all n E Z and z E V? Justify your answer.
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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![Let V = C {z < 0}. Define the principal logarithm Log: V → C as
he branch of log on V such that Log(1) = 0.
(a) Prove that such a branch of log exists on V.
(b) Prove for z E V that Log(z) = ln |z| + iArg(2), where In is the natural logarithm
on positive reals, and Arg is the principal argument with image (-n, 7].
(c) Is Log(zw) = Log(z) + Log(w) for all z, w e V? Justify your answer. Similarly,
is Log(z") = nLog(z) for all n E Z and z E V? Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4d5906d-1721-4e9f-9482-cd9f40e5f11a%2F0e2e52a5-b5fe-4d30-b60a-1b82d5e5da7c%2Fpbkv7bi.png&w=3840&q=75)
Transcribed Image Text:Let V = C {z < 0}. Define the principal logarithm Log: V → C as
he branch of log on V such that Log(1) = 0.
(a) Prove that such a branch of log exists on V.
(b) Prove for z E V that Log(z) = ln |z| + iArg(2), where In is the natural logarithm
on positive reals, and Arg is the principal argument with image (-n, 7].
(c) Is Log(zw) = Log(z) + Log(w) for all z, w e V? Justify your answer. Similarly,
is Log(z") = nLog(z) for all n E Z and z E V? Justify your answer.
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