Residue (complex analysis)

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.

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Contour integration

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. Contour integration methods include: direct integration of a complex-valued function along a curve in the complex plane; application of the Cauchy integral formula; and application of the residue theorem.

Zeros and poles

In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity). Technically, a point z0 is a pole of a function f if it is a zero of the function 1/f and 1/f is holomorphic (i.e. complex differentiable) in some neighbourhood of z0. A function f is meromorphic in an open set U if for every point z of U there is a neighborhood of z in which either f or 1/f is holomorphic.

Residue (complex analysis)

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