Derivative of $f(z)=az$ at infinity is $\frac{1}{a}$? I was going through a dynamical system lecture note and there it is said that the derivative of $f(z)=az$ at infinity is $\frac{1}{a}$, where $z\in \mathbb{C}_\infty$ i.e., Riemann sphere. Normally I would have thought it is $a$.
Could someone please explain why it is $\frac{1}{a}$, and why is the answer $a$ wrong?
Edit: The description in the lecture is:
let $h(z)=\frac{1}{z}$, then $h^{-1}(z)=\frac{1}{z}$. Now let $f_1(z)=h^{-1}\circ f\circ h(z)=\frac{1}{a}z$. Then $f_1(z)^\prime\mid_0=f(z)^\prime\mid_\infty=\frac{1}{a}$. It is the last relation $f_1(z)^\prime\mid_0=f(z)^\prime\mid_\infty=\frac{1}{a}$ which I don't get.
Can a function have more than one derivative at a point?
Thanks a lot.
 A: I would advise you first to track down their definition of "derivative of _ at infinity", because it's not something well-defined.
If $X$ is a one-dimensional complex manifold, $f : X \to X$ is meant to be iterated, and $z \in X$ is a fixpoint of $f$, then if you pick any chart $\phi : (U \subset X) \to (V \subset \Bbb C)$ for some open $U$ containing $z$, you see that the number $\frac {d(\phi \circ f \circ \phi^{-1})}{dx}  (\phi(z))$ doesn't depend on $\phi$ and tells you what happens if you iterate $f$ starting with a point near $z$. If its modulus is less than $1$ then $z$ is called an attractive fixpoint : applying $f$ will attract a point towards $z$ ; if it is greater than $1$, $z$ is called a repulsive fixpoint : it will do the opposite and send the point away from $z$. And the argument tells you by what angle it will turn around $z$ each time.
One of the chart giving complex coordinates near $\infty$ in the Riemann sphere is for example $h : X \setminus \{0\} \to \Bbb C$ given by $x \mapsto 1/x$ and $\infty \mapsto 0$. If you use this chart you are left having to do the calculation described in the lecture.
A: The map $z\mapsto az$ on the Riemann sphere in an infinitesimal neighborhood of the origin is an expansion with factor $a$, while in an infinitesimal neighborhood of the "opposite" point "$\infty$" it is an expansion by $1/a$ (i.e. a contraction by $a$). To formalize this, change the variable by means of $w=1/z$ and then consider what happens at $w=0$.
