Complex Residue

The constant  in the Laurent series

(1)

of  about a point  is called the residue of . If  is analytic at , its residue is zero, but the converse is not always true (for example,  has residue of 0 at  but is not analytic at ). The residue of a function  at a point  may be denoted . The residue is implemented in the Wolfram Language as Residue[f, z, z0].

Two basic examples of residues are given by  and  for .

The residue of a function  around a point  is also defined by

(2)

where  is counterclockwise simple closed contour, small enough to avoid any other poles of . 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  at a point  by writing  in a coordinate  around . Then

(3)

The sum of the residues of  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  may be found without explicitly expanding into a Laurent series as follows. If  has a pole of order  at , then  for  and . Therefore,

(4)

			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

Iterating,

(12)

So

			(13)
			(14)

and the residue is

(15)

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.