spherical derivative

Let G be a domain.


Let f:G be a meromorphic function, then the spherical derivative of f, denoted f is defined as


for z where f(z) and when f(z)= define


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


If f:GC is a meromorphic function then

Note that sometimes the spherical derivative is also denoted as μ(f)(z) rather then f(z).


  • 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
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