# residue at infinity

If in the Laurent expansion

$f(z)={\displaystyle \sum _{k=-\mathrm{\infty}}^{\mathrm{\infty}}}{c}_{k}{z}^{k}$ | (1) |

of the function $f$, the coefficient ${c}_{n}$ is distinct from zero ($n>0$) and ${c}_{n+1}={c}_{n+2}=\mathrm{\dots}=0$, then there exists the numbers $M$ and $K$ such that

$$ |

In this case one says that $\mathrm{\infty}$ is a *pole of order* $n$ of the function $f$ (cf. zeros and poles of rational function).

If there is no such positive integer $n$, (1) infinitely many positive powers of $z$, and one may say that $\mathrm{\infty}$ is an *essential singularity ^{}* of $f$.

In both cases one can define for $f$ the *residue at infinity* as

$\frac{1}{2i\pi}}{\displaystyle {\oint}_{C}}f(z)\mathit{d}z={c}_{-1},$ | (2) |

where the integral is taken along a closed contour $C$ which goes once anticlockwise around the origin, i.e. once clockwise around the point $z=\mathrm{\infty}$ (see the Riemann sphere^{}).

Then the usual form

$$\frac{1}{2i\pi}{\oint}_{C}f(z)\mathit{d}z=\sum _{j}\text{Res}(f;{a}_{j})$$ |

of the residue theorem^{} may be expressed as follows:

*The sum of all residues of an analytic function ^{} having only a finite number of points of singularity is equal to zero.*

## References

- 1 Ernst Lindelöf: Le calcul des résidus et ses applications à la théorie des fonctions. Gauthier-Villars, Paris (1905).

Title | residue at infinity |
---|---|

Canonical name | ResidueAtInfinity |

Date of creation | 2013-03-22 19:15:00 |

Last modified on | 2013-03-22 19:15:00 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 30D30 |

Related topic | Residue |

Related topic | RegularAtInfinity |