Meromorphic function

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In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. (The terminology comes from the Ancient Greek meros (μέρος), meaning part, as opposed to holos (ὅλος), meaning whole.)

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 then, 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 be infinite; if both parts have a zero at z, then one must compare the multiplicities of these zeros.)

From an algebraic point of view, if D 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 \mathbb{Q}, the rational numbers, and \mathbb{Z}, the integers.

Additionally, in group theory of the 1930s, a meromorphic function (or simply a meromorph) was a function from a group G into itself which preserves the product on the group. The image of this function was called an automorphism of G [1]. (Similarly, a homomorphic function (or homomorph) was a function between groups which preserved the product while a homomorphism was the image of a homomorph.) This terminology has been replaced with use of endomorphism for the function itself with no special name given to the image of the function and thus meromorph no longer has an implied meaning within group theory.

Contents

Examples

  • The functions
  • The function
  • 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

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