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2. (Cauchy-Riemann's equations, analyticity and harmonic functions) (a) Define the symbols af /Əz and af /əz by af_1 (af 1 əf' af_1 (af 1ðf -- 2 ar i dy dz 2 ar i dy, as suggested by the relations z = (z + ž), y = ±(2 – E) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to ôf/ðž = 0. Also, show that if f is analytic, then f' = @f /dz. H%3D

2. (Cauchy-Riemann's equations, analyticity and harmonic functions) (a) Define the symbols af /Əz and af /əz by af_1 (af 1 əf' af_1 (af 1ðf -- 2 ar i dy dz 2 ar i dy, as suggested by the relations z = (z + ž), y = ±(2 – E) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to ôf/ðž = 0. Also, show that if f is analytic, then f' = @f /dz. H%3D

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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2. (Cauchy-Riemann's equations, analyticity and harmonic functions) (a) Define the symbols ðf /dž and ôf/dz by af 1 (af laf° af_1 (af ¸ 1 af %3D az 2 ər i ðy, dz 2 ax i ðy as suggested by the relations z = }(z + 2), y = ±(2 – 2) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to ðf/ðž = 0. Also, show that if f is analytic, then f' = ðf /dz. (b) Determine all functions f = u + iv that are analytic in the whole plane and has the property that the real part u is a function of only y = Im z. The answer should be given as an expression in the variable z = x + iy.
Transcribed Image Text:2. (Cauchy-Riemann's equations, analyticity and harmonic functions) (a) Define the symbols ðf /dž and ôf/dz by af 1 (af laf° af_1 (af ¸ 1 af %3D az 2 ər i ðy, dz 2 ax i ðy as suggested by the relations z = }(z + 2), y = ±(2 – 2) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to ðf/ðž = 0. Also, show that if f is analytic, then f' = ðf /dz. (b) Determine all functions f = u + iv that are analytic in the whole plane and has the property that the real part u is a function of only y = Im z. The answer should be given as an expression in the variable z = x + iy.
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