I'm reading Bak & Newman's complex analysis.
And this is the first time to contact with complex analysis.
At chapter 7, after proving Schwarz's lemma, the authors give "magical" bilinear transformation at least to me.
Which is
$B_\alpha(z)=\frac{z-a}{1-\overline{\alpha}z}$ where $|\alpha|<1$.
(They say it is useful to solve some extremal problem. and I'm convinced after examining some examples.)
After searching this transformation for a while, I found this is so called conformal mapping. And in the book which I'm reading, I didn't encounter conformal mapping yet.
But that's all I found, and I'm not sure why this mapping has to be defined as above.
Is there any approachable resource?or could you give the intuition or background of which
why it is defined as above?
Thanks in advance a lot.
In general, on the real line, you can send any three points to any other three points with a transformation of the form $$ x \mapsto \frac{Ax + B}{Cx + D} $$ which is a so-called "projective" transformation (and is undefined when the denominator is zero).
This extends nicely to the complex "line" (i.e., the set of complex numbers).
The transformation you're looking at is one of these. It helps to look at what it does to $z = 0, a,$ and $\infty$: it sends them to $-a, 0, and \frac{a}{\bar{a}}= \frac{a^2}{\bar{a}a} = \frac{a^2}{|a|^2} = \left( \frac{a}{|a|}\right)^2$, which is just a point on the unit circle that's "twice as far around" as $a$ is.
I'm not sure that helps, but at least the transformation arises from a fairly natural context of projective maps.