Residue at infinity (${\mathrm{Res}}_{\infty }f=-[z^{-1}]f$)

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: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.