Separable Function: Alternative Representation

Question

How does one get the following function

$$ f(u) = f(x+iy) = \frac{u^{z-1}}{e^{-u}-1}, $$

where $z$ is a constant complex number and u is a complex variable, into the form:

$$ f(x+iy) = v(x,y) + iw(x,y) $$

?

Best Regards,

J.B

Answer 1

Here are a few basic exercises to get you started.

1. Write $(\alpha+i\beta)\cdot(\gamma+i\delta)$ in the form $\square+i\square$.
2. Write $(\alpha+i\beta)/(\gamma+i\delta)$ in the form $\square+i\square$.
3. Write $e^{\alpha+i\beta}$ in the form $\square+i\square$.
4. Write $\alpha+i\beta$ as $\square e^{i\square}$ (polar form).
5. Write $\log(re^{i\theta})$ as $\square+i\square$.

Once you can do all of these, you can solve your problem. (Since $u^{z-1}=e^{(\log u)(z-1)}$.)