How to determine if an ugly function is harmonic on a domain. Let $\Omega \subseteq \mathbb{R}^2$ be a domain (open and connected set). By the standard definition, a function $u:\Omega \to \mathbb{R}$ is harmonic $\iff u$ is twice continuously differentiable and it satisfies laplace's equation $u_{xx}+u_{yy}=0$ on the domain. This definition works fine for the most part, but what if we have a function like:
 $$u: \Omega \to \mathbb{R},\ (x,y) \mapsto \sin(x^2+y^2+\cos(x^2-y^2)+37).$$
We are too lazy to differentiate this! And we certainly wouldn't want to differentiate it under time constrains like a test. Is there a quick way to determine if this function is harmonic or not? 
Looking through my notes, I found this might be helpful, but I'm not entirely sure how to apply it: If $u$ is entire harmonic, then the image of $u$ is either constant or all of $\mathbb{R}$.

It is helpful. Note that $u(x,y)=\sin(\text{something real})$. So $-1 \leq u(x,y) \leq 1$ for all $x,y$. Now, show that $u$ is non-constant...
