The residue of a meromorphic function at an isolated singularity, often denoted or, is the unique value such that has an analyticantiderivative in a punctured disk. Alternatively, residues can be calculated by finding Laurent series expansions, and one can define the residue as the coefficienta−1 of a Laurent series. The definition of a residue can be generalized to arbitrary Riemann surfaces. Suppose is a 1-form on a Riemann surface. Let be meromorphic at some point, so that we may write inlocal coordinates as. Then the residue of at is defined to be the residue of at the point corresponding to.
As an example, consider the contour integral where C is some simple closed curve about 0. Let us evaluate this integral using a standard convergence result about integration by series. We can substitute the Taylor series for into the integrand. The integral then becomes Let us bring the 1/z5 factor into the series. The contour integral of the series then writes Since the series converges uniformly on the support of the integration path, we are allowed to exchange integration and summation. The series of the path integrals then collapses to a much simpler form because of the previous computation. So now the integral around C of every other term not in the form cz−1 is zero, and the integral is reduced to The value 1/4! is the residue of ez/z5 at z = 0, and is denoted
Calculating residues
Suppose a punctured disk D = in the complex plane is given and f is a holomorphic function defined on D. The residue Res of f at c is the coefficient a−1 of in the Laurent series expansion of f around 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. According to the residue theorem, we have: where γ traces out a circle around c in a counterclockwise manner. We may choose the path γ to be a circle of radius ε around c, where ε is as small as we desire. 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.
Removable singularities
If the function f can be continued to a holomorphic function on the whole disk, then Res = 0. The converse is not generally true.
Simple poles
At a simple polec, the residue of f is given by: It may be that the function f can be expressed as a quotient of two functions,, where g and h are holomorphic functions in a neighbourhood of c, with h = 0 and h' ≠ 0. In such a case, L'Hôpital's rule can be used to simplify the above formula to:
Limit formula for higher order poles
More generally, if c is a pole of order n, then the residue of f around z = c can be found by the formula: 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.
Residue at infinity
In general, the residue at infinity is given by: If the following condition is met: then the residue at infinity can be computed using the following formula: If instead then the residue at infinity is
Series methods
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.
