residue at infinity

If in the Laurent expansion

$f(z)={\displaystyle \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}}{c}_k{z}^k$(1)

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

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

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

${\displaystyle \frac{1}{2i\pi }}{\displaystyle {\oint }_C}f(z)𝑑z=c_{-1},$(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  $z=\mathrm{\infty }$ (see the Riemann sphere).

Then the usual form

$$\frac{1}{2i\pi }{\oint }_{C}f(z)𝑑z=\sum _j\text{Res}(f;a_j)$$

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.