Keyhole contour and it's Example || Contour Integration || Complex Analysis || Math Contributor

This article is about the line integral in the complex plane. For the general line integral, see Line integral.
Part of a series of articles about
Calculus
Fundamental theorem
Leibniz integral rule
Limits of functionsContinuity
Mean value theoremRolle's theorem
Differential
Integral
Lists of integralsIntegral transform
Definitions
AntiderivativeIntegral (improper)Riemann integralLebesgue integrationContour integrationIntegral of inverse functions
Integration by
PartsDiscsCylindrical shellsSubstitution (trigonometric, Weierstrass, Euler)Euler's formulaPartial fractionsChanging orderReduction formulaeDifferentiating under the integral signRisch algorithm
Series
Vector
Multivariable
Specialized
Miscellaneous
vte
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 (a contour);
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.This article is about the line integral in the complex plane. For the general line integral, see Line integral.
Part of a series of articles about
Calculus
Fundamental theorem
Leibniz integral rule
Limits of functionsContinuity
Mean value theoremRolle's theorem
Differential
Integral
Lists of integralsIntegral transform
Definitions
AntiderivativeIntegral (improper)Riemann integralLebesgue integrationContour integrationIntegral of inverse functions
Integration by
PartsDiscsCylindrical shellsSubstitution (trigonometric, Weierstrass, Euler)Euler's formulaPartial fractionsChanging orderReduction formulaeDifferentiating under the integral signRisch algorithm
Series
Vector
Multivariable
Specialized
Miscellaneous
vte
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 (a contour);
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.
Keyhole contour and it's Example || Contour Integration || Complex Analysis || Math Contributor
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