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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
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 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 imaginary parts of f(z), respectively. The partial derivatives Ux vy for the partial derivatives Vy Uy = = 1 2 are continuous in the entire complex Vx = 2(2x − y) · 2 vy= 2(2x − y) ∙ (−1) - plane. Here I am using the symbols ux, uy, Vx and Əv and respectively. The Cauchy-Riemann ду’ Ju du Iv дх’ 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 = Y Zo if and only if zo lies on the line 2x - number t. 12 In other words, ƒ(z) is differentiable at 13 2 1 20 t+ (2t+)i for some real i.e., zo =
Transcribed Image Text: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 imaginary parts of f(z), respectively. The partial derivatives Ux vy for the partial derivatives Vy Uy = = 1 2 are continuous in the entire complex Vx = 2(2x − y) · 2 vy= 2(2x − y) ∙ (−1) - plane. Here I am using the symbols ux, uy, Vx and Əv and respectively. The Cauchy-Riemann ду’ Ju du Iv дх’ 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 = Y Zo if and only if zo lies on the line 2x - number t. 12 In other words, ƒ(z) is differentiable at 13 2 1 20 t+ (2t+)i for some real i.e., zo =
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