If in the Laurent expansion

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

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) contains 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

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

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.