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. The infinity
∞
is a point added to the local space
C
in order to render it compact (in this case it is a one-point compactification). This space noted
C
^
is isomorphic to the Riemann sphere. One can use the residue at infinity to calculate some integrals.
Given a holomorphic function f on an annulus
A
(
0
,
R
,
∞
)
(centered at 0, with inner radius
R
and infinite outer radius), the residue at infinity of the function f can be defined in terms of the usual residue as follows:
R
e
s
(
f
,
∞
)
=
−
R
e
s
(
1
z
2
f
(
1
z
)
,
0
)
Thus, one can transfer the study of
f
(
z
)
at infinity to the study of
f
(
1
/
z
)
at the origin.
Note that
∀
r
>
R
, we have
R
e
s
(
f
,
∞
)
=
−
1
2
π
i
∫
C
(
0
,
r
)
f
(
z
)
d
z
