meromorphic functions of several variables
Definition.
Let be a domain and let be a function. is called if for each there exists a neighbourhood () and twoholomorphic (http://planetmath.org/HolomorphicFunctionsOfSeveralVariables)functions defined in where is not identically zero, such that outside the set where .
Note that is really defined only outside of a complex analytic subvariety. Unlike in one variable, we cannot simply define to be equal to at the poles and expect to be a continuous mapping to some larger space (the Riemann sphere in the case of one variable). The simplest counterexample in is , 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 , and and have no common factors, then the indeterminancy set of is the set where .
References
- 1 Lars Hörmander.,North-Holland Publishing Company, New York, New York, 1973.
- 2 Steven G. Krantz.,AMS Chelsea Publishing, Providence, Rhode Island, 1992.