# Counting Fractional Linear Transformations

This is a problem from Ahlfors' Complex analysis, Section 3.5.

In an obvious way, which we shall not try to make precise, a family of [Möbius] transformations depends on a certain number of real parameters. How many real parameters are there in the family of all [fractional] linear transformations? How many in the families of hyperbolic, elliptic, parabolic transformations? How many fractional linear transformations leave a given circle invariant?

I'm not really sure what he's getting at. I am wondering my thoughts below are in the general direction of what kind of reasoning he's looking for.

All fractional linear transformations

$Tz = \frac{az + b}{cz + d}$ looks like it has 4 complex numbers, so at first I thought there should be 8 real numbers involved. But, many collections give the same transformation, so if you normalize using the determinant, you're left with three degrees of freedom in complex numbers. So you need 6 real parameters.

Hyperbolic transformations

You can write these like $\frac{Tz - a}{Tz - b} = k \frac{z-a}{z-b}$, where $k$ is real, and $a$ and $b$ are complex. So you need 5 real numbers to specify one.

Elliptic transformations

Pretty much the same as hyperbolic, except $k = e^{i\theta}$. So also 5 parameters in this case.

Parabolic transformations

You can write these like $\frac{\omega}{Tz - a} = \frac{\omega}{z-a} + c$ where $\omega$ and $a$ are complex and $c$ is real. So again 5 real parameters.

Transformations taking a given circle to itself

Pich three points on the circle $z_1, z_2, z_3$. Transformations taking the circle to itself are in bijective correspondence with triples $w_1, w_2, w_3$ with $|w_i - c| = r$. So $w_i = c + r e^{ i \theta_i}$. One real parameter is required for each $w_i$, and so three real parameters are required to specify such a transformation.

Your answers are correct, except for the parabolic case. Only four real parameters are needed there: you could divide the equation by $c$, for example.