Mathematical PhysicsComplex analysis 1)The functions u(x, y) and v(x, y) are the real and imaginary parts, respectively, of ananalytic function w(z).(a) Assuming that the required derivatives exist, show thatv²u =v²v= 0.%3D%3DSolutions of Laplace`s equation such as u (x, y) and v(x, y) are called harmonicfunctions.(b) Show thatθυ θυ=0.дх дуди дидх ду2) Find the analytic function w(z) = u(x , y)+i v(x ,y) if u(x ,y) = e* cos y.For f(z) = In z and g(z) =z2, find f'(z) and g'(z).%3DIdentify the maximal region within which these two functions are analytic.

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Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 78E

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