Fractional Linear Transformations; # of Parameters.

Question

A fractional linear transformation is a function $$ w = f(z) = \frac{az+b}{cz+d} $$ where $a,b,c,d$ are complex constants satisfying $ ad-bc \not = 0$.

In the book Complex Analysis, by Gamelin, the author makes the remark (p.64),

A fractional linear transformation depends on 4 complex parameters. One of these can be adjusted without changing the transformation.

Does this make sense? I'm unsure how to understand that statement, for instance, I believe if we decided to change the constant $b$, then it doesn't necessarily hold that $$ \frac{az+b_1}{cz+d} = \frac{az+b_2}{cz+d} \text{ right?}$$

What am I missing? Any explanation would be greatly appreciated!

Thanks :)

Answer 1

I think it means that if $b\neq 0$, you can change $b$ to whatever you want and retain the transformation. However you will have to possibly change $a,b,c$ as well. Assume we want to change $b$ to $B\neq 0$, then: $$ \frac{az+b}{cz+d} = \frac{b}{B}\frac{\frac{B}{b}az+B}{cz+d} = \frac{(\frac{b}{B}a)z+B}{(\frac{B}{b}c)z+(\frac{B}{b}d)}. $$ So you can change $b$ to $B$ if you allow $a,b,c$ to change and so retain the transformation.

Answer 2

What this somewhat-cryptically refers to is that changing all of $a,b,c,d$ by the same non-zero constant does not change the action. That is, while $GL_2(\mathbb C)$ acts by linear fractional transformations, its center (the scalars) act trivially, so the action "descends" to the quotient of $GL_2(\mathbb C)$ by its center, which is called $PGL_2(\mathbb C)$.