(d) Demonstrate mathematical reasoning to prove that that the real-valued function х2 — у? u(х, у) %3D х3 + 2х2 - 3ху? 2у2 + (x² + y2)2 is harmonic. Construct a harmonic conjugate of u(x, y) and write down an entire function f(z) such that Re(f(z)) = u(x,y), where z = x + yi. In your working, you may use the integration formula a? – 3y? dy J (a² + y²)³ y + C (a² + y²)² for any real constant a.

(d) Demonstrate mathematical reasoning to prove that that the real-valued function х2 — у? u(х, у) %3D х3 + 2х2 - 3ху? 2у2 + (x² + y2)2 is harmonic. Construct a harmonic conjugate of u(x, y) and write down an entire function f(z) such that Re(f(z)) = u(x,y), where z = x + yi. In your working, you may use the integration formula a? – 3y? dy J (a² + y²)³ y + C (a² + y²)² for any real constant a.

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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Question

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(d)
Demonstrate mathematical reasoning to prove that that the real-valued function
x2 – y2
u(x, y) = x3 + 2x² – 3xy2 – 2y² +
(x2 + y2)2
is harmonic. Construct a harmonic conjugate of u(x, y) and write down an entire
function f(z) such that Re(f(z)) = u(x, y), where z = x + yi. In your working, you
may use the integration formula
a? – 3y?
y
(a2 + y?)3
dy
(a? + y2)²
+ C
for any real constant a.

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