Contour integration

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${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
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In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.^[1][2][3]

Contour integration is closely related to the calculus of residues,^[4] 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.^[5]

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.

One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums.