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,{r_m,r_m},…]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

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 GenerateConditionsAutomatic:

Use the option GenerateConditionsFalse to suppress the existence conditions:

Generate conditions when the contour involves parameters:

Suppress the conditions using GenerateConditionsFalse:

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: