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(a) Define the symbols ôf/DE and af/dz by af 1 (af 10f 2 (dr af 1 (of 18f i by, dz 2 dr' i dy as suggested by the relations z (:+ =), y = ±(: - =) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to Of/0 = 0. Also, show that if f is analytic, then f'= 0f/0z.

(a) Define the symbols ôf/DE and af/dz by af 1 (af 10f 2 (dr af 1 (of 18f i by, dz 2 dr' i dy as suggested by the relations z (:+ =), y = ±(: - =) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to Of/0 = 0. Also, show that if f is analytic, then f'= 0f/0z.

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 1af dz 2 dx i dy, af _1(af ¸ 1af dz 2 (dr" i dy, as suggested by the relations z = }(2+ 2), y = ±(z – =) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to df/ðz = 0. Also, show that if f 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 = 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 1af dz 2 dx i dy, af _1(af ¸ 1af dz 2 (dr" i dy, as suggested by the relations z = }(2+ 2), y = ±(z – =) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to df/ðz = 0. Also, show that if f 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 = x + iy.
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