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?

  • $\begingroup$ This time you received a good answer from Alexei below, but in the future you will want to present formulae as Mathematica expressions, so people don't have to type them in manually. The less work we have to do, the more likely we are to help :-) $\endgroup$
    – MarcoB
    Commented Oct 19, 2015 at 14:01

1 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!

  • $\begingroup$ Many thanks, exactly what I was looking for. $\endgroup$ Commented Oct 19, 2015 at 22:07

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