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# regular at infinity

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

$\varrho\;<\;|z|\;<\;\infty,$ |

it has a Laurent expansion

$\displaystyle w(z)\;=\;\sum_{{n=-\infty}}^{{\infty}}c_{n}z^{n}.$ | (1) |

If especially the coefficients $c_{1},\,c_{2},\,\ldots$ vanish, then we have

$w(z)\;=\;c_{0}+\frac{c_{{-1}}}{z}+\frac{c_{{-2}}}{z^{2}}+\ldots$ |

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

$w\!\left(\frac{1}{\zeta}\right)\;=\;c_{0}+c_{{-1}}\zeta+c_{{-2}}\zeta^{2}+\ldots$ |

is regular in the disc $|\zeta|<\varrho$. Accordingly we can define that the function $w$ is regular at infinity also.

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

Keywords:

regular

Related:

RegularFunction, ClosedComplexPlane, VanishAtInfinity, ResidueAtInfinity

Synonym:

analytic at infinity

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

30D20*no label found*32A10

*no label found*

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## Comments

## Little suggestion

I would suggest changing

$$w(\frac{1}{\zeta}) = c_0+c_{-1}\zeta+c_{-2}\zeta^2+\ldots$$

to

$$w \left( \frac{1}{\zeta} \right) = c_0 + c_{-1} \zeta + c_{-2} \zeta^2 + \ldots$$

so that the parentheses for the fraction go all the way in both directions, though that would make it look like the Jacobi symbol or something; anyway as for the spacing, it never hurts.

## Re: Little suggestion

Thanks. You are fully right.