Let Log(z) denote the principal branch of the complex logarithm, so that Log(z) = In |z| + i Arg(z), z #0, %3D where In r denotes the usual natural logarithm of the number r > 0. Consider now the formula f(2) = Log(z? + i). (i) For which values of z E C does this formula define a function of complex variable? (ii) Find the largest open set A such that f(2) is holomorphic on A and compute f'(z) for z E A.
Let Log(z) denote the principal branch of the complex logarithm, so that Log(z) = In |z| + i Arg(z), z #0, %3D where In r denotes the usual natural logarithm of the number r > 0. Consider now the formula f(2) = Log(z? + i). (i) For which values of z E C does this formula define a function of complex variable? (ii) Find the largest open set A such that f(2) is holomorphic on A and compute f'(z) for z E A.
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 Log(z) denote the principal branch of the complex logarithm, so that
Log(z) = In |2| + i Arg(z),
z #0,
where In r denotes the usual natural logarithm of the number r > 0.
Consider now the formula
f(2) = Log(2? + i).
(i) For which values of z EC does this formula define a function of complex variable?
(ii) Find the largest open set A such that f(z) is holomorphic on A and compute f'(z) for
Z E A.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fbc9d8fa6-a850-41de-8014-d74cdc162f76%2Fpyzvcgln_processed.png&w=3840&q=75)
Transcribed Image Text:Let Log(z) denote the principal branch of the complex logarithm, so that
Log(z) = In |2| + i Arg(z),
z #0,
where In r denotes the usual natural logarithm of the number r > 0.
Consider now the formula
f(2) = Log(2? + i).
(i) For which values of z EC does this formula define a function of complex variable?
(ii) Find the largest open set A such that f(z) is holomorphic on A and compute f'(z) for
Z E A.
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