spherical derivative
Let $G\subset \u2102$ be a domain.
Definition.
Let $f:G\to \u2102$ be a meromorphic function, then the spherical derivative of $f$, denoted ${f}^{\mathrm{\u266f}}$ is defined as
$${f}^{\mathrm{\u266f}}(z):=\frac{2{f}^{\prime}(z)}{1+{f(z)}^{2}}$$ 
for $z$ where $f(z)\ne \mathrm{\infty}$ and when $f(z)=\mathrm{\infty}$ define
$${f}^{\mathrm{\u266f}}(z)=\underset{\zeta \to z}{lim}{f}^{\mathrm{\u266f}}(\zeta ).$$ 
The second definition makes sense since a meromorphic functions has only isolated poles, and thus ${f}^{\mathrm{\u266f}}(\zeta )$ is defined by the first equation when we are close to $z$. Some basic properties of the spherical derivative are as follows.
Proposition.
If $f\mathrm{:}G\mathrm{\to}\mathrm{C}$ is a meromorphic function then

•
${f}^{\mathrm{\u266f}}$ is a continuous function^{},

•
$$ for all $z\in G$.
Note that sometimes the spherical derivative is also denoted as $\mu (f)(z)$ rather then ${f}^{\mathrm{\u266f}}(z)$.
References
 1 John B. Conway. . SpringerVerlag, New York, New York, 1978.
 2 Theodore B. Gamelin. . SpringerVerlag, New York, New York, 2001.
Title  spherical derivative 

Canonical name  SphericalDerivative 
Date of creation  20130322 14:18:36 
Last modified on  20130322 14:18:36 
Owner  jirka (4157) 
Last modified by  jirka (4157) 
Numerical id  7 
Author  jirka (4157) 
Entry type  Definition 
Classification  msc 30D30 