Question?I understand 2x-y=-½.how to get z0= t+ (2t+½)i?? Problem 6. Let ƒ(z) = x + 2y + (2x − y)² i, where z = x + yi.(a) Find all complex numbers zo such that ƒ(z) is differentiable at zo?(b) For every zo where f(z) is differentiable, find the complex derivative ƒ'(zo).(c) Find all complex numbers zo such that f(z) is analytic at z = zo?Solution: (a) Let u(x, y) = x + 2y and v(x, y) = (2x − y)² be the real and imaginaryparts of f(z), respectively. The partial derivativesUxvy for the partial derivativesVyUy==12are continuous in the entire complexVx = 2(2x − y) · 2vy= 2(2x − y) ∙ (−1)-plane. Here I am using the symbols ux, uy, Vx andƏvand respectively. The Cauchy-Riemannду’Ju du Ivдх’ ду’ дх’equations, Vy =ux and v= -Uy, translate to-2(2x - y) = 1 and 4(2x - y) = −2.Both are satisfied if and only if 2x - y =YZo if and only if zo lies on the line 2x -number t.12In other words, ƒ(z) is differentiable at132120 t+ (2t+)i for some reali.e., zo =

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9) 2₂ (a) Find all complex numbers zo such that f(z) is differentiable at zo? (b) For every zo where f(z) is differentiable, find the complex derivative f'(zo). (c) Find all complex numbers zo such that f(z) is analytic at z zo? Solution: (a) Let u(x, y) = x+2y and v(x, y) = (2x- y)² be the real and imaginary arts of f(z), respectively. The partial derivatives Ux = 1 Uy = 2 Vx = 2(2x - y) 2 vy=2(2x - y) (-1) re continuous in the entire complex plane. Here I am using the symbols ux, Uy, Vx and Əv du du dv for the partial derivatives dx' dy' dx' quations, vy = ux and vx = -Uy, translate to and respectively. The Cauchy-Riemann ду’ -2(2x - y) = 1 and 4(2x - y) Both are satisfied if and only if 2x - y = 5 if and only if zo lies on the line 2x 11/13 2 1 y = = -2. In other words, f(z) is differentiable at 1 i.e., zo = t+ (2t+)i for some real

9) 2₂ (a) Find all complex numbers zo such that f(z) is differentiable at zo? (b) For every zo where f(z) is differentiable, find the complex derivative f'(zo). (c) Find all complex numbers zo such that f(z) is analytic at z zo? Solution: (a) Let u(x, y) = x+2y and v(x, y) = (2x- y)² be the real and imaginary arts of f(z), respectively. The partial derivatives Ux = 1 Uy = 2 Vx = 2(2x - y) 2 vy=2(2x - y) (-1) re continuous in the entire complex plane. Here I am using the symbols ux, Uy, Vx and Əv du du dv for the partial derivatives dx' dy' dx' quations, vy = ux and vx = -Uy, translate to and respectively. The Cauchy-Riemann ду’ -2(2x - y) = 1 and 4(2x - y) Both are satisfied if and only if 2x - y = 5 if and only if zo lies on the line 2x 11/13 2 1 y = = -2. In other words, f(z) is differentiable at 1 i.e., zo = t+ (2t+)i for some real

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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