When finding a harmonic function, from a conformal mapping, is it ok to have imaginary numbers in the argument,

Question

Say after finding a conformal mapping, using this conformal mapping, I find a harmonic function

$$\arg \left[\frac{(z^2+1)}{(z^2-1)}\right]$$

that satisfies boundary conditions that I want.

Is it necessary to expand the $z^2$ and make an attempt to "get rid of" the imaginary numbers?

After applying the $\arg(z)$ function, the result will be real-valued as we want for harmonic functions, but I'm just wondering whether the inputs are allowed to be complex numbers.

(I have tried expanding and simplifying and have not gotten anywhere so far...)

Thanks,

Answer 1

Letting $z = x + iy$, this defines a map $$\Bbb R \times \Bbb R \to \Bbb R : (x, y) \mapsto \arg\left [\frac {(x + iy)^2 + 1}{(x + iy)^2 - 1}\right].$$

It doesn't matter how the map is described. Only that it goes from $\Bbb R^2 \to \Bbb R$ and satisfies the harmonic equation. Real, imaginary and argument functions of holomorphic functions are a well-known source for harmonic functions.