spherical derivative

Let $G\subset ℂ$ be a domain.

Definition.

Let $f:G\to ℂ$ be a meromorphic function, then the spherical derivative of $f$, denoted $f^{\mathrm{♯}}$ is defined as

$$f^{\mathrm{♯}}(z):=\frac{2|f^{\prime }(z)|}{1+{|f(z)|}^2}$$

for $z$ where $f(z)\ne \mathrm{\infty }$ and when $f(z)=\mathrm{\infty }$ define

$$f^{\mathrm{♯}}(z)=\underset{\zeta \to z}{lim}{f}^{\mathrm{♯}}(\zeta ).$$

The second definition makes sense since a meromorphic functions has only isolated poles, and thus $f^{\mathrm{♯}}(\zeta )$ 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

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

• for all $z\in G$.

Note that sometimes the spherical derivative is also denoted as $\mu (f)(z)$ rather then $f^{\mathrm{♯}}(z)$.

Title	spherical derivative
Canonical name	SphericalDerivative
Date of creation	2013-03-22 14:18:36
Last modified on	2013-03-22 14:18:36
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	7
Author	jirka (4157)
Entry type	Definition
Classification	msc 30D30