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Complex Residue

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The constant 👁 a_(-1)
in the Laurent series

of 👁 f(z)
about a point 👁 z_0
is called the residue of 👁 f(z)
. If 👁 f
is analytic at 👁 z_0
, its residue is zero, but the converse is not always true (for example, 👁 1/z^2
has residue of 0 at 👁 z=0
but is not analytic at 👁 z=0
). The residue of a function 👁 f
at a point 👁 z_0
may be denoted 👁 Res_(z=z_0)(f(z))
. The residue is implemented in the Wolfram Language as [f, 👁 {
z, z0👁 }
].

Two basic examples of residues are given by 👁 Res_(z=0)1/z=1
and 👁 Res_(z=0)1/z^n=0
for 👁 n>1
.

The residue of a function 👁 f
around a point 👁 z_0
is also defined by

where 👁 gamma
is counterclockwise simple closed contour, small enough to avoid any other poles of 👁 f
. In fact, any counterclockwise path with contour winding number 1 which does not contain any other poles gives the same result by the Cauchy integral formula. The above diagram shows a suitable contour for which to define the residue of function, where the poles are indicated as black dots.

It is more natural to consider the residue of a meromorphic one-form because it is independent of the choice of coordinate. On a Riemann surface, the residue is defined for a meromorphic one-form 👁 alpha
at a point 👁 p
by writing 👁 alpha=fdz
in a coordinate 👁 z
around 👁 p
. Then

The sum of the residues of 👁 intfdz
is zero on the Riemann sphere. More generally, the sum of the residues of a meromorphic one-form on a compact Riemann surface must be zero.

The residues of a function 👁 f(z)
may be found without explicitly expanding into a Laurent series as follows. If 👁 f(z)
has a pole of order 👁 m
at 👁 z_0
, then 👁 a_n=0
for 👁 n<-m
and 👁 a_(-m)!=0
. Therefore,

Multiplying both sides by 👁 (z-z_0)^m
gives

Take the first derivative and reindex,

Take the second derivative and reindex,

Iterating then gives

So

since 👁 lim_(z->z_0)(z-z_0)^n=0
, and the residue is

The residues of a holomorphic function at its poles characterize a great deal of the structure of a function, appearing for example in the amazing residue theorem of contour integration.