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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.
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Related publications (35)
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Related concepts (16)
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.
Residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. The residue theorem should not be confused with special cases of the generalized Stokes' theorem; however, the latter can be used as an ingredient of its proof.
Essential singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category essential singularity is a "left-over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles. In practice some include non-isolated singularities too; those do not have a residue.