= Residue at infinity
{title2=$\operatorname{Res}_{\infty}f=-[z^{-1}]f$}
For a <function> $f$ holomorphic outside a sufficiently large disk, the residue at infinity is defined using the coordinate $w=1/z$ on the <Riemann sphere>:
$$
\operatorname{Res}_{z=\infty}f(z)\,dz=\operatorname{Res}_{w=0}\left[-w^{-2}f(1/w)\right]\,dw.
$$
The factor $-w^{-2}$ comes from $dz=-w^{-2}dw$. Consequently the residue at infinity is the negative of the coefficient of $z^{-1}$ in the large-$z$ <Laurent expansion>. It is not the residue of the composite function $f(1/w)$ by itself.
For $f(z)=1/z$, the finite residue at zero is one and the residue at infinity is minus one. More generally, a meromorphic differential on the <Riemann sphere> has total residue zero: the finite residues and the residue at infinity cancel. Equivalently, the integral over a large counterclockwise circle is $-2\pi i$ times the residue at infinity. The coordinate change explains the sign in that formula.
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