Meromorphic function

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Categories: Complex analysis | Riemann surfaces | Functions and mappings

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.) Such functions are sometimes said to be regular functions or regular on D.

Every meromorphic function on D can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on D: the poles then occur at the zeroes of the denominator.

Image:Gamma abs2.png

The Gamma function is meromorphic in the whole complex plane

Intuitively then, a meromorphic function is a ratio of two nice (holomorphic) functions. Such a function will still be "nice", except at the points where the denominator of the fraction is zero, when the value of the function will be infinite.

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Q} , the rational numbers, and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z} , the integers.

1 Examples
2 Properties
3 Meromorphic functions on Riemann surfaces
4 Higher dimensions
5 References

Examples

All rational functions such as

Failed to parse (Missing texvc executable; please see math/README to configure.): f(z)= \frac{z^{3}-2z+10}{z^{5}+3z-1} \

are meromorphic on the whole complex plane.

The functions

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and Failed to parse (Missing texvc executable; please see math/README to configure.):  f(z)=\frac{ \sin{z}}{(z-1)^{2}} \

as well as the gamma function and the Riemann zeta function are meromorphic on the whole complex plane.

The function

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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 C\{0}.

The complex logarithm function

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is not meromorphic on the whole complex plane, as it cannot be defined on the whole complex plane less an isolated set of points.

The function

Failed to parse (Missing texvc executable; please see math/README to configure.): f(z) = \frac1{\sin (1/z)}

is not meromorphic in the whole plane, since the point Failed to parse (Missing texvc executable; please see math/README to configure.): z = 0

is an accumulation point of poles and is thus not an isolated singularity. The function

Failed to parse (Missing texvc executable; please see math/README to configure.): f(z) = \sin \frac1z

is not meromorphic either, as it has an essential singularity at 0.

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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By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted, multiplied, and the quotient Failed to parse (Missing texvc executable; please see math/README to configure.): f/g

can be formed unless Failed to parse (Missing texvc executable; please see math/README to configure.):  g(z)=0 
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.

Meromorphic functions on Riemann surfaces

On a Riemann surface every point admits an open neighborhood which is isomorphic 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. (This is a special case of the so-called GAGA principle.)

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 (globally defined) 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.

Higher dimensions

In several complex variables, a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example, f(z₁,z₂)=z₁/z₂ 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 holomorphic function with values in the Riemann sphere: There is a set of "indeterminacy" of codimension two (in the given example this set consists of the origin (0,0)).

Unlike in dimension one, in higher dimensions there do exist complex manifolds on which there are no non-constant meromorphic functions, for example, most complex tori.