# 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.

## References

- 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 |