Consider the following function. u(x, y) = 4xy + x + 2y - 5 (a) Verify that the given function u is harmonic in an appropriate domain D. The function u(x, y) has the following second order partial derivatives. a²u 8x² a²u ay² Thus = a²u a²u + ах2 дуг and the function is harmonic. (b) Find v(x, y), a harmonic conjugate of u. (Give your answer in terms of an arbitrary constant C.) v(x, y) = (c) Form the corresponding analytic function f(z) = u + iv satisfying f(21) = −1 + 5/ f(x + y) =
Consider the following function. u(x, y) = 4xy + x + 2y - 5 (a) Verify that the given function u is harmonic in an appropriate domain D. The function u(x, y) has the following second order partial derivatives. a²u 8x² a²u ay² Thus = a²u a²u + ах2 дуг and the function is harmonic. (b) Find v(x, y), a harmonic conjugate of u. (Give your answer in terms of an arbitrary constant C.) v(x, y) = (c) Form the corresponding analytic function f(z) = u + iv satisfying f(21) = −1 + 5/ f(x + y) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the following function.
u(x, y) = 4xy + x + 2y - 5
(a) Verify that the given function u is harmonic in an appropriate domain D.
The function u(x, y) has the following second order partial derivatives.
a²u
ах2
a²u
ay²
Thus
=
a²u
a²u
+
əx² ay²
and the function is harmonic.
(b) Find v(x, y), a harmonic conjugate of u. (Give your answer in terms of an arbitrary constant C.)
v(x, y) =
(c) Form the corresponding analytic function f(z) = u + iv satisfying f(2)= -1 + 5/
f(x + y) =
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