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Cauchy-Riemann equations In the advanced subject of complex variables, a function typically has the form f(x, y) = u(x, y) + iv(x, y), where u and v are real-valued func- tions and i = v-1 is the imaginary unit. A function f = u + iv is said to be analytic (analogous to differentiable) if it satisfies the Cauchy-Riemann equations: u̟ a. Show that f(x, y) = (x² – y²) + i(2ry) is analytic. b. Show that f(x, y) = x(x² – 3y²) + iy(3x² – y*) is analytic. c. Show that if f = u + iv is analytic, then u, + u, = 0 and = v, and u, = -v,- %3D || = 0. Assume u and v satisfy the conditions in yy Theorem 15.4.

Cauchy-Riemann equations In the advanced subject of complex variables, a function typically has the form f(x, y) = u(x, y) + iv(x, y), where u and v are real-valued func- tions and i = v-1 is the imaginary unit. A function f = u + iv is said to be analytic (analogous to differentiable) if it satisfies the Cauchy-Riemann equations: u̟ a. Show that f(x, y) = (x² – y²) + i(2ry) is analytic. b. Show that f(x, y) = x(x² – 3y²) + iy(3x² – y*) is analytic. c. Show that if f = u + iv is analytic, then u, + u, = 0 and = v, and u, = -v,- %3D || = 0. Assume u and v satisfy the conditions in yy Theorem 15.4.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Cauchy-Riemann equations In the advanced subject of complex variables, a function typically has the form f(x, y) = u(x, y) + iv(x, y), where u and v are real-valued func- tions and i = v-1 is the imaginary unit. A function f = u + iv is said to be analytic (analogous to differentiable) if it satisfies the Cauchy-Riemann equations: u̟ a. Show that f(x, y) = (x² – y²) + i(2ry) is analytic. b. Show that f(x, y) = x(x² – 3y²) + iy(3x² – y*) is analytic. c. Show that if f = u + iv is analytic, then u, + u, = 0 and = v, and u, = -v,- %3D || = 0. Assume u and v satisfy the conditions in yy Theorem 15.4.
Transcribed Image Text:Cauchy-Riemann equations In the advanced subject of complex variables, a function typically has the form f(x, y) = u(x, y) + iv(x, y), where u and v are real-valued func- tions and i = v-1 is the imaginary unit. A function f = u + iv is said to be analytic (analogous to differentiable) if it satisfies the Cauchy-Riemann equations: u̟ a. Show that f(x, y) = (x² – y²) + i(2ry) is analytic. b. Show that f(x, y) = x(x² – 3y²) + iy(3x² – y*) is analytic. c. Show that if f = u + iv is analytic, then u, + u, = 0 and = v, and u, = -v,- %3D || = 0. Assume u and v satisfy the conditions in yy Theorem 15.4.
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