The residue at infinity of a function 
that is analytic for
is the complex residue at 
of the one-form obtained from
by the change of variables 
. Since 
, this gives
 |
(1)
|
The factor 
is essential. The residue at infinity is generally different from the complex
residue of 
alone at 
(Gautam 2020, p. 2).
Equivalently, if 
is the counterclockwise circle 
with 
, then
 |
(2)
|
The minus sign corresponds to a clockwise contour surrounding the finite part of the complex plane. If the exterior Laurent series is
 |
(3)
|
then the residue at infinity is 
. For example,
 |  |  |
(4)
|
 |  |  |
(5)
|
The first function is analytic at infinity when considered as a function
on the Riemann sphere, although its one-form 
has a pole
there.
If 
has only finitely many finite isolated singularities 
, ..., 
and is analytic elsewhere
in the complex plane, the residue
theorem gives
 |
(6)
|
The finite singularities may be poles or essential singularities.
