ContourIntegrate

ContourIntegrate[f,z∈cont]

gives the integral of f along the contour defined by cont in the complex plane.

Details and Options

  • Contour integration is also known as path integration or complex line integration.
  • Contour integrals arose in the study of holomorphic and meromorphic functions in complex analysis, but they are now used in a wide range of applications, including the computation of inverse Laplace transforms and Z transforms, definite integrals and sums, and solutions of partial differential equations.
  • The contour integral of a function along a contour cont is given by:
  • The value of the contour integral is independent of the parametrization, but it does depend on the orientation of the contour cont.
  • The function f is typically a meromorphic function of z, but it can be any piecewise continuous function that is defined in a neighborhood of cont in the complex plane.
  • The contour integral of a meromorphic function along a closed contour cont can be computed using Cauchy's residue theorem.
  • Commonly used closed contours cont include: »
  • {"Hairpin", hl}encircle a half-line hl
    {"UpperSemicircle",ipts,epts}encircle the upper half-plane, including the points ipts and excluding the points epts, all on the real axis
    {"LowerSemicircle",ipts,epts}encircle the lower half-plane, including the points ipts and excluding the points epts, all on the real axis
    {"Dumbbell",pt1,pt2}encircle the capsule given by points pt1 and pt2
  • The complex points are given as {x,y} pairs; complex half-lines are given as HalfLine primitives.
  • A contour cont in can also be specified as a curve region (RegionQ) in .
  • For a parametric contour ParametricRegion[{x[t],y[t]},{{t,a,b}}], the orientation is in the direction of increasing t.
  • Special contours in and their assumed orientation:
  • Line[{p1,p2,…}]from p1 to p2, etc.
    HalfLine[{p1,p1}]from p1 toward p2
    InfiniteLine[{p1,p2}]from p1 toward p2
    Circle[p,…]counterclockwise
  • Area regions such as Polygon can be used and the contours are then taken to be the boundary contours RegionBoundary[Polygon[…]].
  • Special area regions in and their assumed boundary contour orientation:
  • Triangle[{p1,p2,p3}]counterclockwise
    Rectangle[p1,p2]counterclockwise
    RegularPolygon[n,…]counterclockwise
    Polygon[{p1,p2,…}{{q1,q2,…},…}]counterclockwise of the outer contour, clockwise for inner contours
    Disk[p,…]counterclockwise
    Ellipsoid[p,…]counterclockwise
    StadiumShape[{p1,p2},r]counterclockwise
    Annulus[p,{rm,rm},…]counterclockwise for outer contour and clockwise for inner contour
  • ContourIntegrate uses a combination of symbolic and numerical methods when the input involves inexact quantities.
  • The regions in cont may be wrapped with Inactive to prevent autoevaluation.
  • The following options can be given:
  • Assumptions $Assumptionsassumptions to make about parameters
    GenerateConditions Automaticwhether to generate answers that involve conditions on parameters
    PrincipalValue Falsewhether to find Cauchy principal values
    WorkingPrecision Automaticthe precision used in internal computations

Examples

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Basic Examples  (3)

Integrate 1/z along the unit circle in the complex plane:

Integrate a rational function along a circle with center at the origin and radius 2:

Contour integral over a polygonal path:

Scope  (53)

Basic Uses  (5)

Contour integral over a circular path:

Numerical integration:

Contour integral over a polygonal chain in the complex plane:

Contour integrate over a half-disk:

Numerical contour integral:

Contour integral over a parametric contour in the complex plane:

Special Topic: Rational Functions  (13)

Integrate a rational function along a circle:

Integrate a parametric rational function along a circle:

Integrate a rational function along a pentagonal contour:

Contour integral of a rational function along a triangular path:

Contour integral of a rational function along a rectangular path:

Contour integral along the unit circle:

Contour integral over an open polygonal chain:

Contour integral over an open arc:

Contour integral of a meromorphic function over a closed semicircle:

Contour integral of a function with an essential singularity:

Contour integral of a non-analytic function:

Contour integral of a function containing a branch cut:

Contour integral of a function depending on a symbolic parameter:

Special Topic: Meromorphic Functions  (5)

Contour integral of a meromorphic function along a polygonal path:

Evaluate the contour integral numerically:

Contour integral along an elliptical path:

Contour integral over a closed semicircle:

Contour integral over a sector of annulus:

Contour integral over a circle of radius :

Special Topic: Functions with Essential Singularities  (4)

Exponential function:

Sin function with an essential singularity inside the contour:

Contour integral of a function with an essential singularity:

Essential singularity arising from a periodic function:

Special Topic: Non-analytic Functions  (4)

Contour integral over a circular path:

Contour integral of the Arg function:

Contour integral over an elliptic sector:

Contour integral over a rectangular path:

Special Topic: Functions with Branch Cuts  (2)

Contour integral of a piecewise continuous function:

Contour integral of a function with branch cuts on the integration path:

Special Topic: Symbolic Parameters  (5)

Function and contour can contain symbolic parameters:

Suppress conditions of existence using GenerateConditions False:

Result is a Piecewise function:

Contour integral along a half-disk of radius :

Contour integral along a generic ellipse:

Contour integral along a half-annulus of radii and :

Special Topic: Named Contours  (8)

Contour integral along the real axis in positive direction, around poles on the real axis, closing in the upper half of the complex plane:

A second example:

Contour integral along the real axis in positive direction, around poles on the real axis, closing in the lower half of the complex plane:

By default, this contour is traversed clockwise.

A second example:

Contour integral around a hairpin or Hankel contour:

Integral around a hairpin or Hankel contour:

For the case :

Contour integral that evaluates to a Zeta function:

For the case , :

Contour integral on a hairpin:

For the case :

Hairpin or Hankel contour:

Dumbbell contour around the branch cut joining 0 and 1:

Special Topic: Region Contours  (7)

Contour integral over an infinite line:

For the case :

Contour integral over a circular contour:

Contour integral over a line segment:

Contour integral over a triangular path:

For the case :

Contour integral over a rectangular path:

Contour integral over a sector:

Contour integral over an annulus:

Options  (6)

Assumptions  (1)

The option Assumptions can be used on parameters:

GenerateConditions  (2)

ContourIntegrate generates conditions on parameters with GenerateConditionsAutomatic:

Use the option GenerateConditionsFalse to suppress the existence conditions:

Generate conditions when the contour involves parameters:

Suppress the conditions using GenerateConditionsFalse:

PrincipalValue  (1)

The option PrincipalValue can be used to calculate the Cauchy principal value:

WorkingPrecision  (2)

When WorkingPrecision is set, the integral is evaluated numerically:

If the input has a finite precision, the integral is evaluated numerically:

Applications  (22)

Rational Functions  (2)

Contour integral on a half-disk:

Limit for large :

The same result obtained with Integrate:

Integral over the real line:

This can be obtained as the limit of a contour integral:

Trig-Rational Products  (2)

Integrals on the real line:

This result can be recovered using a complex integral:

Integrals on the real line:

Use a complex integral:

Trigonometric Functions  (3)

Integral of a rational function of the sine:

This can be recovered as a contour integral:

Integral of a rational function of the cosine:

This can be obtained as a contour integral:

Integral of a rational function of the sine:

As a contour integral:

Fourier Transform  (2)

Fourier transform of a function:

Computation using a contour integral: for positive :

For negative :

Fourier transform of a function:

Computation using a contour integral: for positive :

For negative :

Inverse Laplace Transform  (4)

Inverse Laplace transform of a function:

Computation using a contour integral:

Inverse Laplace transform of a rational function:

Using a contour integral:

Inverse Laplace transform of a function containing a square root:

The same computation using a contour integral:

Inverse Laplace transform of a function containing Log:

Use the definition of the inverse Laplace transform:

Inverse Mellin Transform  (4)

Inverse Mellin transform of a function:

Compute it from a contour integral:

Inverse Mellin transform of a function:

Compute it from its definition as a contour integral:

Mellin transform of a function:

Recover the function using an inverse Mellin transform:

Mellin transform of a rational function:

Relation to the inverse Mellin transform:

Inverse Z Transform  (2)

Inverse Z transform of a function:

Obtain the result from its definition as a contour integral:

Inverse Z transform of a function:

Use its definition as a contour integral:

Classical Theorems  (3)

Residue theorem applied to the contour integral of a meromorphic function over a closed path:

The integral is equal to times the sum of the residues of the poles inside the contour:

The integration contour can be deformed without changing the value of the integral, provided that no singularities of the function are crossed:

If no singularities lie inside the contour, the integral is zero:

Properties & Relations  (6)

Apply N[ContourIntegrate[...]] to obtain a numerical solution if the symbolic calculation fails:

This can also be computed using NIntegrate:

Contour integrals can also be obtained using Integrate:

This is equivalent to:

Integrate can integrate along a straight contour in the complex plane:

This is equivalent to:

Contour integrals over a closed path can also be obtained using ResidueSum:

Poles of a meromorphic function can be found using FunctionPoles:

The integral can also be computed using Residue:

Contour integrals over a closed path can also be obtained using Residue:

Interactive Examples  (2)

Contour integral over a sector of varying radius:

Another contour integral over a sector of varying radius:

Neat Examples  (2)

Contour integral over a "Pacman" contour:

Contour integral of a meromorphic function along a "Ninja" contour:

Wolfram Research (2023), ContourIntegrate, Wolfram Language function, https://reference.wolfram.com/language/ref/ContourIntegrate.html.

Text

Wolfram Research (2023), ContourIntegrate, Wolfram Language function, https://reference.wolfram.com/language/ref/ContourIntegrate.html.

CMS

Wolfram Language. 2023. "ContourIntegrate." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ContourIntegrate.html.

APA

Wolfram Language. (2023). ContourIntegrate. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ContourIntegrate.html

BibTeX

@misc{reference.wolfram_2024_contourintegrate, author="Wolfram Research", title="{ContourIntegrate}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/ContourIntegrate.html}", note=[Accessed: 27-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_contourintegrate, organization={Wolfram Research}, title={ContourIntegrate}, year={2023}, url={https://reference.wolfram.com/language/ref/ContourIntegrate.html}, note=[Accessed: 27-April-2024 ]}