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A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. Every meromorphic function can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0): the poles then occur at the zeroes of the denominator.
Examples of meromorphic functions are all rational functions such as f(z) = (z3 − 2z + 1)/(z5 + 3z − 1), the functions f(z) = exp(z)/z and f(z) = sin(z)/(z − 1)2 as well as the gamma function and the Riemann zeta function. The functions f(z) = ln(z) and f(z) = exp(1/z) are not meromorphic.
By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted, multiplied, and the quotient f/g can be formed unless g(z) = 0 for all z. Thus, the meromorphic functions form a field, in fact a field extension of the complex numbers.
In the language of Riemann surfaces, a meromorphic function is the same as a holomorphic function that maps to the Riemann sphere and which is not constant ∞. The poles correspond to those complex numbers which are mapped to ∞.
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