simplifiy complex expression

Question

How to simplify the following expression:

$$\frac{z-1}{z+1}~, \quad \text{where} z\in \mathbb{C}\setminus \{-1\}$$

There is just nothing i can come up with, neither in cartesian, nor in polar.

Answer 1

Your expression is about as simplified as it can get. That being the case, I assume that when you say "simplify" you mean that you wish to evaluate the expression and need to get rid of the complex denominator. To this end:

Let...

$$f(z)=\frac{z-1}{z+1}$$

Let $v=z-1$, and $w=z+1$. Then...

$$f(z)=\frac{v}{w}=\frac{\bar{w}v}{|w|^2}$$

Substitute $z-1$ and $z+1$ back in for $v$ and $w$, respectively...

$$f(z)=\frac{(\bar{z}+1)(z-1)}{|z+1|^2}$$

The denominator is now a real number, and you may proceed as usual from here.