Contour integration

This article is about the line integral in the complex plane. For the general line integral, see Line integral.

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 used to study complex-valued functions that are holomorphic in a region.

Contour integration is closely related to the calculus of residues, a method of complex analysis. The power of contour integration comes from the fact that the integrals of holmorphic functions are invariant under deforming the contour, provided the deformation does not cross a singularity or branch cut. Thus the value of a contour integral between fixed endpoints is not governed by the precise shape of the contour, but by its winding around the singularities of the integrand.

One use of contour integrals is the evaluation of certain integrals of functions over the real line. Regarding the real line as a contour, and deforming it into the complex plane often leads to simpler integrals than those which can be found by using only real variable methods. In modern language, the integral of a holomorphic or meromorphic function is a pairing between a cohomology class of differential forms and a homology class of cycles in the domain of the function. It also has various applications to physics.

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