meromorphic functions of several variables


Let Ωn be a domain and let h:Ω be a function. h is called if for each pΩ there exists a neighbourhood UΩ (pU) and two holomorphic ( functions f,g defined in U where g is not identically zero, such that h=f/g outside the set where g=0.

Note that h is really defined only outside of a complex analytic subvariety. Unlike in one variable, we cannot simply define h to be equal to at the poles and expect h to be a continuous mapping to some larger space (the Riemann sphere in the case of one variable). The simplest counterexampleMathworldPlanetmath in 2 is (z,w)z/w, which does not have a unique limit at the origin. The set of points where there is no unique limit, is called the indeterminancy set. That is, the set of points where if h=f/g, and f and g have no common factors, then the indeterminancy set of h is the set where f=g=0.


  • 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
  • 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title meromorphic functions of several variables
Canonical name MeromorphicFunctionsOfSeveralVariables
Date of creation 2013-03-22 16:01:10
Last modified on 2013-03-22 16:01:10
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 4
Author jirka (4157)
Entry type Definition
Classification msc 32A20
Defines indeterminancy set