Кешендік жазықтықтағы мероморфтық функциялар

Class of mathematical function
In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a set of isolated points, which are poles of the function. The term comes from the Greek meros (μέρος), meaning "part". Every meromorphic function on D can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on D: any pole must coincide with a zero of the denominator.

Intuitively, a meromorphic function is a ratio of two well behaved (holomorphic) 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 at z 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.

Both the field of study wherein the term is used and the precise meaning of the term changed in the 20th century. In the 1930s, in group theory, a meromorphic function (or meromorph) 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 (or homomorph) 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. A meromorphic function is not necessarily an endomorphism, since the complex points at its poles are not in its domain, but may be in its range.

Since poles are isolated, there are at most countably many for a meromorphic function. for example are meromorphic on the whole complex plane. Furthermore, they are the only meromorphic functions on the extended complex plane. The functions as well as the gamma function and the Riemann zeta function are meromorphic on the whole complex plane. The function is defined in the whole complex plane except for the origin, 0. However, 0 is not a pole of this function, rather an essential singularity. Thus, this function is not meromorphic in the whole complex plane. However, it is meromorphic (even holomorphic) on The complex logarithm function is not meromorphic on the whole complex plane, as it cannot be defined on the whole complex plane while only excluding a set of isolated points. The function is not meromorphic in the whole plane, since the point is an accumulation point of poles and is thus not an isolated singularity. The function is not meromorphic either, as it has an essential singularity at 0.