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A meromorphic function is a complex function defined on an open subset of the complex plane which is holomorphic on all except of a set of isolated points, which are poles for the function. Such functions are sometimes also called regular functions or regular on D.
Every meromorphic function on can be expressed as the ratio of two holomorphic functions defined on with the denominator not identically equal 0. Clearly, the poles then occur at the zeroes of the denominator.
The set of poles of an meromorphic function can be infinite. Take e.g. . Since the poles of a meromorphic function are isolated, there are at most countably many with a possible accumulation point at the complex point at infinity.
Gamma function is a function meromorphic on the whole complex plane. But the complex logarithm is not meromorphic there, because it cannot be defined on the whole complex plane except an isolated set of points.
The name comes from the Ancient Greek meros meaning part, as opposed to holos meaning whole.
If is also connected (to be able to use analytic continuation), then the set of meromorphic functions on form the field of fractions of the integral domain of the set of holomorphic functions on . This field is actually a field extension of the field of complex numbers .
Cite this web-page as:
Štefan Porubský: Meromorphic functions.