How to solve this question on harmonic functions?

Question

How can I determine $a$ so that the given function is harmonic, and find its harmonic conjugate?

$$u = e^{\pi x}\cos(av)$$

Where $v$ is itself a real valued function of x,y.

Is there any other method than using Laplace Equation and taking double derivative and solving the equation as it tends to become too complicated?

There must be some hypothesis on $v$; if $v = y$, $a = \pi$ makes $u$ harmonic, but if $v = x$ there is no solution for $a$. — Robert Lewis, Aug 16 '17 at 19:01
And in light of my last comment, $v$ harmonic is not the right assumption. — Robert Lewis, Aug 16 '17 at 19:03

Erock Brox · Accepted Answer · 2017-08-16 19:05:50Z

4

Here is my attempt.

Use definition of Laplace equation. Solutions of Laplace's equation are harmonic functions.

$f(x,v)=e^{\pi x}cos(av)$

$\Delta f=\frac{\partial^2f }{\partial x^2}+\frac{\partial^2f }{\partial v^2}=0$

After taking second partials and plugging them into Laplace we get:

$\pi ^{2}cos(av)e^{\pi x}-a^{2}e^{\pi x}cos(av)=0$

$e^{\pi x}cos(av)(\pi ^{2}-a^{2})=0$

$a=\pm \pi$

answered Aug 16 '17 at 19:05

Erock Brox

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$\begingroup$ Your answer is correct but are you considering v to be independent of x? $\endgroup$ – Sherry Aug 17 '17 at 16:31
1

$\begingroup$ Yes, its the only way that I can make sense of the problem. $\endgroup$ – Erock Brox Aug 18 '17 at 18:07

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