Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y) = 2x(1 – y); (b) u(x, y) = 2x - x3 + 3xy2;

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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EXERCISES
Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y)
when
(a) u(x, y) = 2x(1 – y);
(c) u(x, y) = sinh x sin y;
(b) u(x, y) = 2xr - x³ + 3xy2:
(d) u(x, y) = y/(x? + y²).
Ans. (a) v(x, y) = x2 – y2 + 2y;
(c) v(x, y) = - cosh x cos y;
(b) v(x, y) = 2y - 3x²y+ y³;
(d) v(x, y) = x/(x² + y?).
Verify that the following functions u are harmonic, and in each case give a conjugate
harmonic function v (i.e. v such that u+iv is analytic).
(a) u(x,y) = 3x?y+2x² – y³ – 2y²,
(b) u(x,y) = In(x² +y²).
Transcribed Image Text:EXERCISES Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y) = 2x(1 – y); (c) u(x, y) = sinh x sin y; (b) u(x, y) = 2xr - x³ + 3xy2: (d) u(x, y) = y/(x? + y²). Ans. (a) v(x, y) = x2 – y2 + 2y; (c) v(x, y) = - cosh x cos y; (b) v(x, y) = 2y - 3x²y+ y³; (d) v(x, y) = x/(x² + y?). Verify that the following functions u are harmonic, and in each case give a conjugate harmonic function v (i.e. v such that u+iv is analytic). (a) u(x,y) = 3x?y+2x² – y³ – 2y², (b) u(x,y) = In(x² +y²).
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