Meromorphic functions (nonfiction)
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 [[ Zeros and poles (nonfiction)|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.
Prior, alternate use
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.
Properties
Since poles are isolated, there are at most countably many for a meromorphic function. The set of poles can be infinite ...
By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted, multiplied ...
... 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.
Consider 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.
Examples
All rational functions are meromorphic on the whole complex plane. Furthermore, they are the only meromorphic functions on the extended complex plane (see Reimann sphere (nonfiction)).
In the News
Fiction cross-reference
Categories
Nonfiction cross-reference
- Functions (nonfiction)
- Gnomon Chronicles (nonfiction)
- Elliptical functions (nonfiction)
- Cousin problems (nonfiction)
- Mittag-Leffler's theorem (nonfiction)
- Weierstrass factorization theorem (nonfiction)
Categories
External links
- Meromorphic functions @ Wikipedia
Social media
- [ Post] @ Twitter (25 November 2025)
