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.

 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