Simplifying an expression with complex variables

Question

The other day I was trying to simplify the following expression of $w \in \mathbb{C}$ and $|w| < 1$:

$$f(w) = \left|\frac{2}{(\frac{w+1}{1-w}+1)^2}\right|^{-\Delta} \cdot \frac{1}{\left|{\rm Re}(\frac{w+1}{-w+1})\right|^{\Delta}}$$ for any $\Delta > 0$

It was not obvious (to me) that this is invariance under $w\rightarrow w\,e^{i\alpha}\quad\forall \alpha \in \mathbb{R}$; or rotationally invariant.

Indeed, after a few lines of algebra it simplifies to:

$$f(w) = \frac{2^{\Delta}}{(1-|w|^2)^{\Delta}}$$

My question is: how could have I asked this simplification of Mathematica; i.e., how would I tell Mathematica to replace an expression of a complex variable by radial coordinates and then apply every possible simplification?

Answer 1

Try this:

f = Abs[2/((w + 1)/(1 - w) + 1)^2]^-d*1/Abs[Re[(w + 1)/(1 - w)]]^d

Simplify[ComplexExpand[f /. {w -> w0*Exp[I*a]}], {w0 > 0, d > 0, a > 0}]

(* Out: 2^d Abs[-1 + w0^2]^-d   *)

Have fun!