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19.18. Show that each of the following functions-call each one u(x,y)– is harmonic, and find the function's harmonic partner, v(x, y), such that u(x, y) + iv(x, y) is analytic. Hint: Use C-R conditions. (a) x³ – 3ry?. (b) eª cos y. (c) x² + y² where x² + y² 0.

19.18. Show that each of the following functions-call each one u(x,y)– is harmonic, and find the function's harmonic partner, v(x, y), such that u(x, y) + iv(x, y) is analytic. Hint: Use C-R conditions. (a) x³ – 3ry?. (b) eª cos y. (c) x² + y² where x² + y² 0.

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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My question is about Complex Derivative and Integral. I showed in the upload photo.

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19.18. Show that each of the following functions call each one u(x, y)- is harmonic, and find the function's harmonic partner, v(x, y), such that u(x, y) + iv(x, y) is analytic. Hint: Use C-R conditions. (a) a³ – 3ry?. (b) е" cos y. (c) x2 + y? where x2 + y? 0. -2y (d) е cos 2x. (e) ey²-x² cos 2xy. (f) e" (x cos y – y sin y) + 2 sinh y sin x + x° – 3xy² + y.
Transcribed Image Text:19.18. Show that each of the following functions call each one u(x, y)- is harmonic, and find the function's harmonic partner, v(x, y), such that u(x, y) + iv(x, y) is analytic. Hint: Use C-R conditions. (a) a³ – 3ry?. (b) е" cos y. (c) x2 + y? where x2 + y? 0. -2y (d) е cos 2x. (e) ey²-x² cos 2xy. (f) e" (x cos y – y sin y) + 2 sinh y sin x + x° – 3xy² + y.
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