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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
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.
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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