spherical derivative

Let $G\subset ℂ$ be a domain.

Definition.

Let $f:G\to ℂ$ be a meromorphic function, then the sphericalderivative 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 hasonly 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 derivativeare as follows.

Proposition.

If $f\mathrm{:}G\mathrm{\to }\mathrm{C}$ is a meromorphic functionthen

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

• for all $z\in G$.

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