Intuitively, a meromorphic function is a ratio of two well-behaved functions. Such a function will still be well-behaved, except possibly at the points where the denominator of the fraction is zero. If the denominator has a zero atz and the numerator does not, then the value of the function will approach infinity; if both parts have a zero at z, then one must compare the multiplicity of these zeros. From an algebraic point of view, if the function's domain is connected, then the set of meromorphic functions is the field of fractions of the integral domain of the set of holomorphic functions. This is analogous to the relationship between the rational numbers and the integers.
Prior, alternate use
The field of study where the term is used and the specific meaning of the term changed in the 20th century. In the 1930s, in group theory, a meromorphic function was a function from a group G into itself that preserved the product on the group. The image of this function was called an automorphism of G. Similarly, a homomorphic function was a function between groups that preserved the product, while a homomorphism was the image of a homomorph. This form of the term is now obsolete, and the related term meromorph is no longer used in group theory. The term endomorphism is now used for the function itself, with no special name given to the image of the function.
Properties
Since the poles of a meromorphic function are isolated, there are at most countably many. The set of poles can be infinite, as exemplified by the function By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted, multiplied, and the quotient can be formed unless on a connected component of D. Thus, if D is connected, the meromorphic functions form a field, in fact a field extension of the complex numbers.
Higher dimensions
In several complex variables, a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example, is a meromorphic function on the two-dimensional complex affine space. Here it is no longer true that every meromorphic function can be regarded as a holomorphic function with values in the Riemann sphere: There is a set of "indeterminacy" of codimension two. Unlike in dimension one, in higher dimensions there do exist compact complex manifolds on which there are no non-constant meromorphic functions, for example, most complex tori.
On a Riemann surface, every point admits an open neighborhood which is biholomorphic to an open subset of the complex plane. Thereby the notion of a meromorphic function can be defined for every Riemann surface. When D is the entire Riemann sphere, the field of meromorphic functions is simply the field of rational functions in one variable over the complex field, since one can prove that any meromorphic function on the sphere is rational. For every Riemann surface, 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 ∞. On a non-compact Riemann surface, every meromorphic function can be realized as a quotient of two holomorphic functions. In contrast, on a compact Riemann surface, every holomorphic function is constant, while there always exist non-constant meromorphic functions. Meromorphic functions on an elliptic curve are also known as elliptic functions.
