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Let u (x,y) = e ² cosy + x² - y² 2 u is harmonic Find all v (xay) such that u(xsy)tiv (xsy) is aralytic on C C. Use couchy Reimann equations e cos y + 2x X Vy = 0x = vx = - 4y = ex siny +2y v (x₂y) = Se * siny + 2y dx =e*siny + 2xy + C(y) = v(x,y) and To Find C (y) use Vy = e X Vý = e c'y C(y) is constant C so v(x₂y) = ex siny + 2xy+CA C is any (ع) cosy c'(y) +2x x cos y - +2x+ constant

Let u (x,y) = e ² cosy + x² - y² 2 u is harmonic Find all v (xay) such that u(xsy)tiv (xsy) is aralytic on C C. Use couchy Reimann equations e cos y + 2x X Vy = 0x = vx = - 4y = ex siny +2y v (x₂y) = Se * siny + 2y dx =e*siny + 2xy + C(y) = v(x,y) and To Find C (y) use Vy = e X Vý = e c'y C(y) is constant C so v(x₂y) = ex siny + 2xy+CA C is any (ع) cosy c'(y) +2x x cos y - +2x+ constant

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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Let u(x,y)=e* cosy + x² - y² is harmonic Find all v (xy) such that u (xsy) +iv (xsy) is aralytic on C. couchy Reimann equations X Vy = Ux = e cos y+ 2x √x = - 4y = ex siny +2y v (x₂y) = Se * siny + 2y dx Use =e*siny + 2xy + C(y) = v(x,y) use To Find C (y) X e = vy: Vý = e c'(y) so X cosy cos y =0 and + 2x + c²y +2x → C(y) is constant C siny +2 хутс Cis v(x,y)=@ Ang any constant
Transcribed Image Text:Let u(x,y)=e* cosy + x² - y² is harmonic Find all v (xy) such that u (xsy) +iv (xsy) is aralytic on C. couchy Reimann equations X Vy = Ux = e cos y+ 2x √x = - 4y = ex siny +2y v (x₂y) = Se * siny + 2y dx Use =e*siny + 2xy + C(y) = v(x,y) use To Find C (y) X e = vy: Vý = e c'(y) so X cosy cos y =0 and + 2x + c²y +2x → C(y) is constant C siny +2 хутс Cis v(x,y)=@ Ang any constant
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