Complex Functions Analysis $f(z)=\frac{z-a}{1-\bar{a}z}$

Here’s a question I’m working on:

let $a,z \in \mathbb{C}$, $f(z)=\frac{z-a}{1-\bar{a}z}$

If $|z| = 1$, find $|z-a|^2 – |1-\bar{a}z|^2$

Here’s my progress:
let $z = x+iy$
$a = b+ic$
$\Rightarrow$ $|z-a|^2 – |1-\bar{a}z|^2 = (x-b)^2 – (y-c)^2 – (1-bx-cy)^2 – (cx – by)^2$

But I don’t know how to use the fact that $|z|=1$ or if I could use the function given somehow, and I’m not sure exactly what the question is asking. Am I missing something here?

Anyway, thanks.

Solutions Collecting From Web of "Complex Functions Analysis $f(z)=\frac{z-a}{1-\bar{a}z}$"

Splitting into real and imaginary parts is not generally helpful.

Use
$$|z-a|^2 = (z-a)(\overline{z-a}) = |z|^2 – z\bar a – \bar z a + |a|^2$$
and
$$|1 – \bar a z|^2 = (1 – \bar a z)(\overline{1 – \bar a z}) = 1 – z\bar a – \bar z a + |z|^2|a|^2.$$

What happens if $|z| = 1$?

A much better idea is to use the polar form: if $\lvert z \rvert=1$, then there is a real $\theta$ so $z=e^{i\theta}$. Then
$$ \lvert z-a \rvert = \lvert e^{i\theta}-a \rvert = \lvert e^{i\theta}(1-ae^{-i\theta}) \rvert = \lvert 1-ae^{-i\theta} \rvert = \lvert 1-\bar{a}e^{i\theta} \rvert, $$
using $\lvert z w \rvert = \lvert z \rvert \lvert w \rvert $ and $\lvert z \rvert = \lvert \bar{z} \rvert$.

Following the lead of @Chappers, we let $z=e^{i\theta}$. In addition, let $a=\hat ae^{i\theta}$, where $\hat a =|a|$. Then

$$
|z-a|=|e^{i\theta}-\hat ae^{i\theta}|=|1-\hat a|\\
|1-\bar az|=|1-\hat ae^{-i\theta}e^{i\theta}|=|1-\hat a|\\
|z-a|^2 – |1-\bar{a}z|^2=0
$$

Given $|z|^2=1 \iff z \bar z = 1 \iff \bar z = \cfrac{1}{z}\,$:

$$|z-a|^2 – |1-\bar{a}z|^2=|z-a|^2-|z|^2\cdot\left|\frac{1}{z}-\bar a\right|^2=|z-a|^2 – |\bar z – \bar a|^2=|z-a|^2 – \left|\overline{z – a}\right|^2=0$$