# Fractional Linear Transformations; # of Parameters.

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 :)

• I suppose by "adjusted " here the author means that for any $w$, you can find one transformation that has $a=w$ or $b=w$ ect. Commented Feb 8 at 23:47

## 2 Answers

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 $$PGL_2(\mathbb C)$$.

• Ah, yes, thanks, @TorstenSchoeneberg! :) Commented Mar 27 at 22:51

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.