# residue at infinity

If in the Laurent expansion

 $\displaystyle f(z)\;=\;\sum_{k=-\infty}^{\infty}c_{k}z^{k}$ (1)

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

 $|z^{-n}f(z)|\;<\;M\quad\mbox{always when}\quad|z|\;>\;K.$

In this case one says that $\infty$ 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 $\infty$ is an essential singularity of $f$.

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

 $\displaystyle\frac{1}{2i\pi}\!\oint_{C}\!f(z)\,dz\;=\;c_{-1},$ (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  $z=\infty$ (see the Riemann sphere).

Then the usual form

 $\frac{1}{2i\pi}\!\oint_{C}\!f(z)\,dz\;=\;\sum_{j}\mbox{Res}(f;\,a_{j})$

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 ResidueAtInfinity 2013-03-22 19:15:00 2013-03-22 19:15:00 pahio (2872) pahio (2872) 9 pahio (2872) Definition msc 30D30 Residue RegularAtInfinity