# 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 set of poles. It is also valid the converse

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

Proof. Suppose that  $z\mapsto w(z)$  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 point which is neither a point of regularity (http://planetmath.org/Holomorphic) nor a pole. Let $b_{1},\,b_{2},\,\ldots,\,b_{k}$ and possibly $\infty$ be the poles of the function $w$.

For every  $i=1,\,2,\,\ldots,\,k$,  the function has at the pole $b_{i}$ with the order $n_{i}$, the Laurent expansion of the form

 $\displaystyle w(z)=\frac{c_{-n_{i}}}{(z-b_{i})^{n_{i}}}+\frac{c_{-n_{i}+1}}{(z% -b_{i})^{n_{i}-1}}+\ldots+c_{0}+c_{1}(z-b_{i})+\ldots$ (1)

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

 $\displaystyle w(z)=F_{n_{i}}\!\left(\frac{1}{z-b_{i}}\right)+P(z-b_{i}),$ (2)

where the first addend is the principal part of (1), i.e. consists of the terms of (1) which become infinite in  $z=b_{i}$.

If we think a circle having center in the origin and containing all the finite poles $b_{i}$ (an annulus$\varrho<|z|<\infty$), then $w(z)$ has outside it the Laurent series expansion

 $w(z)=d_{m}z^{m}+d_{m-1}z^{m-1}+\ldots+d_{0}+\frac{d_{-1}}{z}+\ldots,$

which we write, corresponding to (2), as

 $\displaystyle w(z)=G_{m}(z)+Q\!\left(\frac{1}{z}\right)\!,$ (3)

where $G_{m}(z)$ is a polynomial of $z$ and $Q\left(\frac{1}{z}\right)$ a power series in $\frac{1}{z}$. Then the equation

 $R(z)\,:=\sum_{i=1}^{k}F_{n_{i}}\!\left(\frac{1}{z-b_{i}}\right)+G_{m}(z)$

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

 $f(z)\,:=\,w(z)-R(z)$

is analytic (http://planetmath.org/Analytic) everywhere except possibly at the points  $z=b_{i}$  and  $z=\infty$.  If we write

 $f(z)=\left[w(z)-F_{n_{i}}\!\left(\frac{1}{z-b_{i}}\right)\right]-\sum_{j\neq i% }F_{n_{j}}\!\left(\frac{1}{z-b_{j}}\right)-G_{m}(z),$

we see that $f(z)$ is bounded in a neighbourhood of the point $b_{i}$ and is analytic also in this point ($i=1,\,2,\,\ldots,\,k$). But then again, the

 $f(z)=\left[w(z)-G_{m}(z)\right]-\sum_{j=1}^{k}F_{n_{j}}\!\left(\frac{1}{z-b_{j% }}\right)$

shows that $f$ is analytic in the infinity (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.

## References

• 1 R. Nevanlinna & V. Paatero: Funktioteoria.  Kustannusosakeyhtiö Otava, Helsinki (1963).
Title when all singularities are poles WhenAllSingularitiesArePoles 2014-11-21 21:30:22 2014-11-21 21:30:22 pahio (2872) pahio (2872) 17 pahio (2872) Theorem msc 30D10 msc 30C15 msc 30A99 RiemannSphere ZeroesOfAnalyticFunctionsAreIsolated Meromorphic