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2. (Cauchy-Riemann's equations, analyticity and harmonic functions) (a) Define the symbols Of/0E and af/0z by af 1 (af 1af 器-(-)-(+) DE 2 dr i d , Sei Je)i Je 2 (Ori dy as suggested by the relations z = (2+ E), y = ±(z - 3) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to df/У = 0. Also, show that if f is analytic, then f'- aj/0z. (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 2. The answer should be given as an expression in the variable z-z+ iy.

2. (Cauchy-Riemann's equations, analyticity and harmonic functions) (a) Define the symbols Of/0E and af/0z by af 1 (af 1af 器-(-)-(+) DE 2 dr i d , Sei Je)i Je 2 (Ori dy as suggested by the relations z = (2+ E), y = ±(z - 3) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to df/У = 0. Also, show that if f is analytic, then f'- aj/0z. (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 2. The answer should be given as an expression in the variable z-z+ iy.

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/О and df/dz by af 1 (af 1 əf dE 2 dr i dy, af_1 (af ¸ 18af az 2 dx * i dy as suggested by the relations z = (2+ E), y = ±(z – 2) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to ðf/ðž = 0. Also, show that if ƒ is analytic, then f' = df/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 = z + iy.
Transcribed Image Text:2. (Cauchy-Riemann's equations, analyticity and harmonic functions) (a) Define the symbols ðf/О and df/dz by af 1 (af 1 əf dE 2 dr i dy, af_1 (af ¸ 18af az 2 dx * i dy as suggested by the relations z = (2+ E), y = ±(z – 2) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to ðf/ðž = 0. Also, show that if ƒ is analytic, then f' = df/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 = z + iy.
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