# spherical derivative

Let $G\subset{\mathbb{C}}$ be a domain.

###### Definition.

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

 $f^{\sharp}(z):=\frac{2\lvert f^{\prime}(z)\rvert}{1+\lvert f(z)\rvert^{2}}$

for $z$ where $f(z)\not=\infty$ and when $f(z)=\infty$ define

 $f^{\sharp}(z)=\lim_{\zeta\to z}f^{\sharp}(\zeta).$

The second definition makes sense since a meromorphic functions has only isolated poles, and thus $f^{\sharp}(\zeta)$ is defined by the first equation when we are close to $z$. Some basic properties of the spherical derivative are as follows.

###### Proposition.

If $f\colon G\to{\mathbb{C}}$ is a meromorphic function then

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

• $f^{\sharp}(z)<\infty$ for all $z\in G$.

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

## References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
• 2 Theodore B. Gamelin. . Springer-Verlag, New York, New York, 2001.
Title spherical derivative SphericalDerivative 2013-03-22 14:18:36 2013-03-22 14:18:36 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 30D30