Complex Residue

The complex residue of a function at an isolated singularity is the coefficient in the Laurent series

(1)

of about . 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 an integer .

The definition applies to an isolated singularity that is an essential singularity as well as a pole. For example,

			(2)
			(3)
			(4)
			(5)

The last two residues are the coefficients of in the Taylor series of and , respectively. A branch point does not have an ordinary residue because the chosen branch is not a single-valued function that is analytic throughout a deleted neighborhood of that point.

Residues at higher-order poles also occur in products of special functions. For example, has a pole of order 2 at , with

(6)

where is the gamma function and is the Euler-Mascheroni constant.

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

(7)

where is a counterclockwise simple closed contour enclosing and no other singularity, and is holomorphic on and inside the contour except at . More generally, the same result holds for a closed contour with contour winding number 1 about and 0 about every other singularity, under the hypotheses of the residue theorem. The above diagram shows a suitable contour for which to define the residue of a 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 chart. On a Riemann surface, the residue is defined for a meromorphic one-form at a point by writing in a coordinate chart around . Then

(8)

The residue at infinity is obtained by transforming the one-form under . The sum of the residues of a meromorphic one-form 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,

			(9)
			(10)

Multiplying both sides by gives

(11)

Take the first derivative and reindex,

			(12)
			(13)
			(14)

Take the second derivative and reindex,

			(15)
			(16)
			(17)

Iterating then gives

			(18)
			(19)

So

			(20)
			(21)

since , and the residue is

(22)

For a pole of order at , write , so is analytic near and . Its logarithmic derivative provides an alternative formula,

(23)

where is the complete Bell polynomial. For , the argument list is empty and . For example, for the formula is

(24)

Formulas of this type reduce higher-order residue calculations to derivatives of (Liu 2001, p. 134).

The residues of a meromorphic 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.