coefficient of the term of order −1 in the Laurent expansion of a function holomorphic outside a point, whose value can be extracted by a contour integral
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Also known as complex residue
In mathematics, more specifically complex analysis, the residue of a function at a point of its domain is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. Residues are typically readily computed and, once known, allow the determination of general contour integrals via the residue theorem.
In mathematics, more specifically complex analysis, the residue of a function at a point of its domain is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function
that is holomorphic except at the discrete points
{
a
k
}
k
{\displaystyle \{a_{k}\}_{k}}
, which may include essential singularities.) Residues are typically readily computed and, once known, allow the determination of general contour integrals via the residue theorem.
If parts or all of a function can be expanded into a Taylor series or Laurent series, which may be possible if the parts or the whole of the function has a standard series expansion, then calculating the residue is significantly simpler than by other methods. The residue of the function is simply given by the coefficient of
(
z
−
c
)
−
1
{\displaystyle (z-c)^{-1}}
in the Laurent series expansion of the function.
Examples
Residue from series expansion
As an example, consider the contour integral
∮
C
e
z
z
5
d
z
{\displaystyle \oint _{C}{e^{z} \over z^{5}}\,dz}
where
C
{\displaystyle C}
is some simple closed curve about
0
{\displaystyle 0}
.
Let us evaluate this integral using a standard convergence result about integration by series. Substituting the Taylor series for
e
z
{\displaystyle e^{z}}
into the integrand, the integral becomes
∮
C
1
z
5
(
1
+
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has an analytic antiderivative in a punctured disk
0
<
|
z
−
a
|
<
δ
{\displaystyle 0<\vert z-a\vert <\delta }
.
Alternatively, residues can be calculated by finding Laurent series expansions, and one can define the residue as the coefficient
a
−
1
{\displaystyle a_{-1}}
of a Laurent series.
The concept can be used to provide contour integration values of certain contour integral problems considered in the residue theorem. According to the residue theorem, for a meromorphic function
are all isolated singularities within the contour
C
{\displaystyle C}
.
of
f
{\displaystyle f}
at
c
{\displaystyle c}
is the coefficient
a
−
1
{\displaystyle a_{-1}}
of
(
z
−
c
)
−
1
{\displaystyle (z-c)^{-1}}
in the Laurent series expansion of
f
{\displaystyle f}
around
c
{\displaystyle c}
. Various methods exist for calculating this value, and the choice of which method to use depends on the function in question, and on the nature of the singularity.
in a counterclockwise manner and does not pass through or contain other singularities within it. We may choose the path
γ
{\displaystyle \gamma }
to be a circle of radius
ε
{\displaystyle \varepsilon }
around
c
{\displaystyle c}
. Since
ε
{\displaystyle \varepsilon }
can be as small as we desire it can be made to contain only the singularity of
c
{\displaystyle c}
due to nature of isolated singularities. This may be used for calculation in cases where the integral can be calculated directly, but it is usually the case that residues are used to simplify calculation of integrals, and not the other way around.
f
{\displaystyle f}
instead has an essential singularity at
c
{\displaystyle c}
. If the limit is
0
{\displaystyle 0}
, then
f
{\displaystyle f}
is either analytic at
c
{\displaystyle c}
or has a removable singularity there. If the limit is equal to infinity, then the order of the pole is higher than
1
{\displaystyle 1}
.
It may be that the function
f
{\displaystyle f}
can be expressed as a quotient of two functions,
f
(
z
)
=
g
(
z
)
/
h
(
z
)
{\displaystyle f(z)={g(z)}/{h(z)}}
, where
g
{\displaystyle g}
and
h
{\displaystyle h}
are holomorphic functions in a neighbourhood of
c
{\displaystyle c}
, with
h
(
c
)
=
0
{\displaystyle h(c)=0}
and
h
′
(
c
)
≠
0
{\displaystyle h'(c)\neq 0}
. In such a case, L'Hôpital's rule can be used to simplify the above formula to:
This formula can be very useful in determining the residues for low-order poles. For higher-order poles, the calculations can become unmanageable, and series expansion is usually easier. For essential singularities, no such simple formula exists, and residues must usually be taken directly from series expansions.
∞
f
(
z
)
=
0
,
{\displaystyle \lim _{|z|\to \infty }f(z)=0,}
then the residue at infinity can be computed using the following formula:
For functions that are meromorphic on the entire complex plane with finitely many singularities, the sum of the residues at the (necessarily) isolated singularities plus the residue at infinity is zero, which gives: