e to the i Theta, multiplying by a means rotating by Theta and

stretching or shrinking by a factor of absolute value of a.

The second type is a translation, z maps to z + b.

And the third type is an inversion, z maps to 1 over z.

And those are the three types that make up every Möbius transformation.

Let's see why that is true.

Suppose f is a Möbius transformation.

Suppose first that f maps infinity to infinity.

We saw that then f is of the type az + b.

But az + b is a composition of two types of Mobius transformations.

First of all, z gets mapped to az which is the rotation&dilation type.

And then we add b to az which is a translation.

So, az + b is indeed a composition of these two

Möbius transformations the rotation and the translation.

Next, suppose that f of infinity is not equal to infinity,

then f is truly of the form, az + b over cz + d where c is non-zero,

and we can divide all the constants by c.

Therefore, f is of the form a / cz + b / c divided by z + d / c.

In other words, we can make the constant in front of z equal to 1, and

with that constant being equal to 1, we can simply assume that c = 1.

So f(z) is really of the form az + b over z + d, with a 1 in front of the z.