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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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 22E
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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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