What is a harmonic complex function?

Question

So, as far as I have learned, a complex function $f(u)$ is considered harmonic if and only if it satisfies the undermentioned equation:
$$ \frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = 0 $$
for any given complex number $x+iy$.

Is that right ?

Answer 1

I think that a real function $u(x,y)$ is harmonic if it obeys that equation. If it does, then there is another real function $v(x,y)$ that is also harmonic, and there is a complex function $f(x+iy)=u(x,y)+iv(x,y)$ which is differentiable. By that, I mean, you can write $f(x+iy)=g(x+iy,x-iy)=g(z,\overline{z})$, and $\partial g/\partial\overline{z}=0$

Answer 2

Yes, the harmonic functions are those satisying the Laplace equation $\Delta u=0$, where $\Delta\equiv\partial_x^2+\partial_y^2$ is the Laplace operator. Usually one assumes them to be of class $C^2$ (defined on some open subset of the complex plane, say, and taking real values; of course one can consider more general situations), but since any harmonic function admits (locally, which is enough of course) a harmonic conjugate, they are automatically of class $C^\infty$.