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Concept
Residue (complex analysis)
Summary
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function that is holomorphic except at the discrete points {ak}k, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.
The residue of a meromorphic function at an isolated singularity , often denoted , , or , is the unique value such that has an analytic antiderivative in a punctured disk .
Alternatively, residues can be calculated by finding Laurent series expansions, and one can define the residue as the coefficient a−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 in local coordinates as . Then, the residue of at is defined to be the residue of at the point corresponding to .
Computing the residue of a monomial
makes most residue computations easy to do. Since path integral computations are homotopy invariant, we will let be the circle with radius . Then, using the change of coordinates we find that
hence our integral now reads as
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.
Official source
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