In complex analysis, a branch of mathematics, the residue at infinity is a residue of a holomorphic function on an annulus having an infinite external radius.
is a point added to the local space
in order to render it compact (in this case it is a one-point compactification).
This space denoted
is isomorphic to the Riemann sphere.
[1] One can use the residue at infinity to calculate some integrals.
Given a holomorphic function f on an annulus
(centered at 0, with inner radius
and infinite outer radius), the residue at infinity of the function f can be defined in terms of the usual residue as follows: Thus, one can transfer the study of
at infinity to the study of
, we have Since, for holomorphic functions the sum of the residues at the isolated singularities plus the residue at infinity is zero, it can be expressed as:
Res ( f ( z ) , ∞ ) = −
Res
{\displaystyle \operatorname {Res} (f(z),\infty )=-\sum _{k}\operatorname {Res} \left(f\left(z\right),a_{k}\right).}
One might first guess that the definition of the residue of
at infinity should just be the residue of
is that one does not take residues of functions, but of differential forms, i.e. the residue of
at infinity is the residue of