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