How to do this transformation of complex rational function?

Question

Here is a description of constructing Möbius map sending $1$, $i$, $-1$ to 0, 1, $\infty$, respectively

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

I do not know how it was converted to :

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

TIA

Answer 1

When you want to simplify a ratio of two complex numbers such as

$$\frac{a+bi}{c+di}$$

you should multiply the top and bottom by the complex conjugate of the denominator (this is a complex analogue of rationalising the denominator). By doing that we get

$$\frac{a+bi}{c+di}\frac{c-di}{c-di} = \frac{(a+bi)(c-di)}{c^2+d^2}$$

which can then be written in the form $x + yi$ for some real $x$ and $y$.

If you do this for the expression

$$\frac{1+i}{1-i}$$

you will find that it reduces to $-i$.