Part of the proof that shows linear fractional transformations are analytic, is that they are a multiplication of two analytic functions (not including $x = -d/c$):
$$\frac{az+b}{cz+d} = \frac{1}{cz+d} (az+b)$$
I am not convinced (or at least it's not immediately obvious) why the second part is analytic. I can modify the function as follows:
$$\frac{1}{c/d((c/d)x + 1)} = \frac{1}{(1 - (\frac{c}{d}x))} \frac{d}{c}$$.
And the last part can be expanded as a series $\sum{((-c/d) x)^n}$. If I can show it is a power series expansion, is that enough to show the function is analytic?