Meromorphic function

From Free net encyclopedia

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.

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 abs.png

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. That is an analogue to <math> \mathbb{Q}</math> and <math> \mathbb{Z}</math>.

Examples

• All rational functions such as

f(z) = (z³ − 2z + 1)/(z⁵ + 3z − 1)

are meromorphic on the whole complex plane.

• The functions

f(z) = exp(z)/z and f(z) = 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

f(z) = exp(1/z)

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 function f(z) = ln(z) is not meromorphic on the whole complex plane, as it cannot be defined on the whole complex plane except an isolated set of points.

Properties

Since the poles of a meromorphic function are isolated, they are at most countably many. The set of poles can be infinite, as exemplified by the function

f(z)=1/sin(z).

By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted, multiplied, and the quotient f/g can be formed unless 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.

When D is the entire Riemann sphere, this field 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 (more precisely, a meromorphic function on a sphere is a meromorphic function on the plane whose singularity at infinity is either removable or a pole).

In the language of Riemann surfaces, 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 ∞.

References

• Serge Lang, Complex Analysis, Springer, 2003. ISBN 0387985921.

cs:Meromorfní_funkce de:Meromorph fr:Fonction_méromorphe he:פונקציה_מרומורפית it:Funzione_meromorfa ja:有理型関数 pl:Funkcja meromorficzna sl:Meromorfna_funkcija