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i) Find all values of z such that 25 = -1+ v3i. Plot your solutions on a complex plane. ii) For a complex function f(2) = u(x, y) + iv(x, y) with u and v real, the Cauchy-Riemann conditions state that f(2) is analytic if and only if dv du and ду du dx dy Let fi(2) and f2(2) be two analytic functions: prove that the function g(2) defined by g(2) = fi(2)f2(2) is also an analytic function. iii) Using the result of part (ii), or otherwise, determine whether the function f(z) = ze is analytic.

i) Find all values of z such that 25 = -1+ v3i. Plot your solutions on a complex plane. ii) For a complex function f(2) = u(x, y) + iv(x, y) with u and v real, the Cauchy-Riemann conditions state that f(2) is analytic if and only if dv du and ду du dx dy Let fi(2) and f2(2) be two analytic functions: prove that the function g(2) defined by g(2) = fi(2)f2(2) is also an analytic function. iii) Using the result of part (ii), or otherwise, determine whether the function f(z) = ze is analytic.

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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i) Find all values of z such that 25 = -1+ v3i. Plot your solutions on a complex plane. ii) For a complex function f(2) = u(x, y) + iv(x, y) with u and v real, the Cauchy-Riemann conditions state that f(2) is analytic if and only if dv du and ду du dx dy Let fi(2) and f2(2) be two analytic functions: prove that the function g(2) defined by g(2) = fi(2)f2(2) is also an analytic function. iii) Using the result of part (ii), or otherwise, determine whether the function f(z) = ze is analytic.
Transcribed Image Text:i) Find all values of z such that 25 = -1+ v3i. Plot your solutions on a complex plane. ii) For a complex function f(2) = u(x, y) + iv(x, y) with u and v real, the Cauchy-Riemann conditions state that f(2) is analytic if and only if dv du and ду du dx dy Let fi(2) and f2(2) be two analytic functions: prove that the function g(2) defined by g(2) = fi(2)f2(2) is also an analytic function. iii) Using the result of part (ii), or otherwise, determine whether the function f(z) = ze is analytic.
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