Contour integration

Contour integration integrates a complex-valued function along an oriented curve in the complex plane. Its value depends on the function and the curve, and residues often make closed-contour integrals computable.

Complex line integral: An integral of a complex-valued function along a parametrized curve in the complex plane. Contour integration is the curve-based form of this broader integral.
Holomorphic function: A complex function that is complex-differentiable throughout an open domain. Holomorphicity is the central regularity condition behind Cauchy's integral theorem.
Real integral: An integral whose variable and integrand are real-valued. A real integral can sometimes be evaluated by embedding it in a complex contour.
Contour deformation: A change of integration path that preserves an integral under suitable analytic conditions. It changes the contour strategically while retaining the integral's value.
Parametrization: A function that describes a curve by assigning points to parameter values. It turns integration along a contour into an ordinary parameter integral.
Complex plane: The plane whose points represent complex numbers. Contours are oriented curves drawn in this plane.
Fourier transform: A transform that represents a function in terms of its frequency components. Contour deformation evaluates transforms and inverse transforms in many cases.
Real analysis: The branch of mathematics concerned with real numbers, real-valued functions, and their limits and integrals. Real-variable techniques offer an alternative when no useful complex contour exists.
Residue theorem: A theorem equating a closed-contour integral to a weighted sum of enclosed residues. It evaluates many contour integrals from the singularities enclosed by the curve.
Piecewise smooth curve: A curve composed of finitely many smooth segments. This common contour type supports the usual definition of a contour integral.

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