How to determine if the sums and products of harmonic functions is also harmonic?

Question

Suppose I know that $u + iv$ is non-constant and analytic on a domain $D$, then I know that $u$ and $v$ are harmonic on $D$ and not both constant; but how do I then determine whether $3u^2v - v^3 + uv$ (say) is also harmonic on $D$?

Answer 1

The easy way to do this is to recognize the expression.

Let $f(z) = f(x+iy) = u(x,y)+iv(x,y)$ and define $$ g(z) = z^3 + \frac12 z^2 = (x+iy)^3 + \frac12(x+iy)^2 = x^3 - 3xy^2 + \frac12(x^2-y^2) + i(3xy^2 - y^3 + xy). $$

In other words, $\operatorname{Im} g(f(z)) = 3uv^2 - v^3 + uv$ is the imaginary part of an analytic function, hence harmonic.

The difficult way is to compute the Laplacian and use Cauchy-Riemann's equations to simplify the mess.