when all singularities are poles

In the parent entry (http://planetmath.org/ZerosAndPolesOfRationalFunction) we see that a rational function has as its only singularities a finite setMathworldPlanetmath of poles. It is also valid the converseMathworldPlanetmath

Theorem. Any single-valued analytic functionMathworldPlanetmath, which has in the whole closed complex plane no other singularities than poles, is a rational function.

Proof. Suppose that  zw(z)  is such an analytic function. The number of the poles of w must be finite, since otherwise the set of the poles would have in the closed complex plane an accumulation pointMathworldPlanetmathPlanetmath which is neither a point of regularity (http://planetmath.org/Holomorphic) nor a pole. Let b1,b2,,bk and possibly be the poles of the function w.

For every  i=1, 2,,k,  the function has at the pole bi with the order ni, the Laurent expansion of the form

w(z)=c-ni(z-bi)ni+c-ni+1(z-bi)ni-1++c0+c1(z-bi)+ (1)

This is in in the greatest open disc containing no other poles. We write (1) as

w(z)=Fni(1z-bi)+P(z-bi), (2)

where the first addend is the principal part of (1), i.e. consists of the terms of (1) which become infiniteMathworldPlanetmathPlanetmath in  z=bi.

If we think a circle having center in the origin and containing all the finite poles bi (an annulusPlanetmathPlanetmathϱ<|z|<), then w(z) has outside it the Laurent series expansion


which we write, corresponding to (2), as

w(z)=Gm(z)+Q(1z), (3)

where Gm(z) is a polynomialPlanetmathPlanetmath of z and Q(1z) a power seriesMathworldPlanetmath in 1z. Then the equation


defines a rational function having the same poles as w. Therefore the function defined by


is analytic (http://planetmath.org/Analytic) everywhere except possibly at the points  z=bi  and  z=.  If we write


we see that f(z) is boundedPlanetmathPlanetmathPlanetmath in a neighbourhood of the point bi and is analytic also in this point (i=1, 2,,k). But then again, the


shows that f is analytic in the infinityMathworldPlanetmath (http://planetmath.org/RiemannSphere), too. Thus f is analytic in the whole closed complex plane. By Liouville’s theorem, f is a constant function.  We conclude that  R(z)+f(z)=w(z)  is a rational function. Q.E.D.

The theorem implies, that if a meromorphic function is regular at infinity or has there a pole, then it is a rational function.


  • 1 R. Nevanlinna & V. Paatero: Funktioteoria.  Kustannusosakeyhtiö Otava, Helsinki (1963).
Title when all singularities are poles
Canonical name WhenAllSingularitiesArePoles
Date of creation 2014-11-21 21:30:22
Last modified on 2014-11-21 21:30:22
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Theorem
Classification msc 30D10
Classification msc 30C15
Classification msc 30A99
Related topic RiemannSphere
Related topic ZeroesOfAnalyticFunctionsAreIsolated
Related topic Meromorphic