# regular at infinity

When the function $w$ of one complex variable is regular in the annulus

$$ |

it has a Laurent expansion

$w(z)={\displaystyle \sum _{n=-\mathrm{\infty}}^{\mathrm{\infty}}}{c}_{n}{z}^{n}.$ | (1) |

If especially the coefficients ${c}_{1},{c}_{2},\mathrm{\dots}$ vanish, then we have

$$w(z)={c}_{0}+\frac{{c}_{-1}}{z}+\frac{{c}_{-2}}{{z}^{2}}+\mathrm{\dots}$$ |

Using the inversion^{} $z=\frac{1}{\zeta}$, we see that the function

$$w\left(\frac{1}{\zeta}\right)={c}_{0}+{c}_{-1}\zeta +{c}_{-2}{\zeta}^{2}+\mathrm{\dots}$$ |

is regular in the disc $$. Accordingly we can define that the function $w$ is regular at infinity also.

For example, $w(z):={\displaystyle \frac{1}{z}}$ is regular at the point $z=\mathrm{\infty}$ and $w(\mathrm{\infty})=0$. Similarly, ${e}^{\frac{1}{z}}$ is regular at $\mathrm{\infty}$ and has there the value 1.

Title | regular at infinity |

Canonical name | RegularAtInfinity |

Date of creation | 2013-03-22 17:37:30 |

Last modified on | 2013-03-22 17:37:30 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 30D20 |

Classification | msc 32A10 |

Synonym | analytic at infinity |

Related topic | RegularFunction |

Related topic | ClosedComplexPlane |

Related topic | VanishAtInfinity |

Related topic | ResidueAtInfinity |