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

Theorem. 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 $b_1,b_2,\mathrm{\dots },b_k$ and possibly $\mathrm{\infty }$ be the poles of the function $w$.

For every  $i=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots },k$,  the function has at the pole $b_i$ with the order $n_i$, the Laurent expansion of the form

$w(z)={\displaystyle \frac{c_{-n_i}}{{(z-b_i)}^{n_i}}}+{\displaystyle \frac{c_{-n_i+1}}{{(z-b_i)}^{n_i-1}}}+\mathrm{\dots }+c_0+c_1(z-b_i)+\mathrm{\dots }$ (1)

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

$w(z)=F_{n_i}\left({\displaystyle \frac{1}{z-b_i}}\right)+P(z-b_i),$ (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  ), then $w(z)$ has outside it the Laurent series expansion

$$w(z)=d_m{z}^m+d_{m-1}{z}^{m-1}+\mathrm{\dots }+d_0+\frac{d_{-1}}{z}+\mathrm{\dots },$$

which we write, corresponding to (2), as

$w(z)=G_m(z)+Q\left({\displaystyle \frac{1}{z}}\right),$ (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  $z=\mathrm{\infty }$.  If we write

$$f(z)=\left[w(z)-F_{n_i}\left(\frac{1}{z-b_i}\right)\right]-\sum _{j\ne i}{F}_{n_j}\left(\frac{1}{z-b_j}\right)-G_m(z),$$

we see that $f(z)$ is bounded in a neighbourhood of the point $b_i$ and is analytic also in this point ($i=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots },k$). 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.

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