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)

Multiplying both sides by gives

(6)

Take the first derivative and reindex,

			(7)
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			(9)

Take the second derivative and reindex,

			(10)
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			(12)

Iterating then gives

			(13)
			(14)

So

			(15)
			(16)

since , and the residue is

(17)

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.