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