Meromorphy

Meromorphy is the property of a function being meromorphic. A function defined on an open subset of the complex plane is meromorphic if it is holomorphic there except at isolated points where it has poles. Equivalently, near a pole z0 the function has a Laurent expansion with finitely many negative-power terms. If there are no poles in the domain, the function is holomorphic there; if the domain is all of C, such a function is called entire.

Meromorphic functions generalize holomorphic functions by allowing controlled singularities. Near each pole, the function behaves like a finite principal part of a Laurent series, and away from poles it is holomorphic. The value infinity is treated as a permissible value at poles, especially when considering the Riemann sphere.

Examples and context. Rational functions, ratios of polynomials, are meromorphic on the extended complex plane and have poles where the denominator vanishes. The function tan z = sin z / cos z is meromorphic on C, with simple poles at z = pi/2 + k pi. Entire functions such as e^z or sin z are also meromorphic, since they have no poles; however, tan z is not entire due to its poles. On the Riemann sphere, a function is meromorphic if and only if it is rational.

Further, on compact Riemann surfaces, meromorphic functions are holomorphic maps to the Riemann sphere, with finitely many poles. The set of all global meromorphic functions on a given surface forms a function field of transcendence degree one, and the sum, product, and quotient (where defined) of meromorphic functions are also meromorphic.