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*) = (*z*^{3} − 2*z* + 1)/(*z*^{5} + 3*z* − 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 to the Riemann sphere which is not constant ∞. The poles correspond to those complex numbers which are mapped to ∞. |