# NContourIntegrate

NContourIntegrate[f,zcont]

gives the numerical 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 orientations:
•  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 .
• Special area regions in and their assumed boundary contour orientations:
•  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
• The regions in cont may be wrapped with Inactive to prevent auto-evaluation.
• The following options can be given:
•  AccuracyGoal Automatic digits of absolute accuracy sought MaxPoints Automatic maximum total number of sample points MaxRecursion Automatic maximum number of recursive subdivisions Method Automatic method to use MinRecursion 0 minimum number of recursive subdivisions PrecisionGoal Automatic digits of precision sought WorkingPrecision Automatic the precision used in internal computations

# Examples

open allclose all

## Basic Examples(3)

Integrate 1/z along the unit circle:

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

Integrate a meromorphic function along an elliptical contour:

Compare the result with ContourIntegrate:

## Scope(46)

### Basic Uses(9)

Contour integral over a circular path:

Compare to ContourIntegrate:

Contour integral over a polygonal chain in the complex plane:

Contour integrate over a half-disk:

Numerical contour integral of a trigonometric expression:

Contour integral over a parametric contour in the complex plane:

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:

### Special Topic: Rational Functions(8)

Integrate a 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 rational function along a circular path:

### Special Topic: Meromorphic Functions(5)

Contour integral of a meromorphic function along a polygonal path:

Evaluate the contour integral symbolically:

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 6:

### 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: Named Contours(7)

Contour integral along the real axis in a 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 a 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:

Compare to the symbolic evaluation:

Contour integral that evaluates to a Zeta function:

Numerical evaluation:

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:

Contour integral over a circular contour:

Contour integral over a line segment:

Contour integral over a triangular path:

Contour integral over a rectangular path:

Contour integral over a sector:

Contour integral over an annulus:

## Options(7)

### AccuracyGoal(1)

The option AccuracyGoal sets the number of digits of accuracy:

The result with default settings only sets a PrecisionGoal:

### MaxPoints(1)

The option MaxPoints stops the integration after a specified number of points has been evaluated:

### MaxRecursion(1)

The option MaxRecursion specifies the maximum number of recursive steps:

Increasing the number of recursions:

The exact result is:

### Method(1)

The option Method can take the same values as in NIntegrate. For example:

With the default option:

Compare to the truncated exact result:

### MinRecursion(1)

The option MinRecursion forces a minimum number of subdivisions:

Compare to the exact result:

### PrecisionGoal(1)

The option PrecisionGoal sets the relative tolerance in the integration:

With default settings:

### WorkingPrecision(1)

Using WorkingPrecision, the working precision can be set:

## Applications(22)

### Rational Functions(2)

Contour integral on a half-disk of large radius:

It agrees with the limit for large computed symbolically:

The same result obtained with NIntegrate:

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:

These two results can be recovered using a complex integral along a half-disk of large radius:

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:

For positive :

Computation using a numerical contour integral:

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:

For :

Computation using a contour integral:

Inverse Laplace transform of a logarithm of a rational function:

For :

Using a contour integral:

Inverse Laplace transform of a function containing a square root:

For :

The same computation using a contour integral:

Inverse Laplace transform of a function containing Log:

For :

Use the definition of the inverse Laplace transform:

### Inverse Mellin Transform(4)

Inverse Mellin transform of a function:

For :

Compute it from a contour integral:

Inverse Mellin transform of a function:

For :

Compute it from its definition as a contour integral:

Mellin transform of a function:

Recover the function at using an inverse Mellin transform:

This is the same as:

Mellin transform of a rational function:

Recover the function at using an inverse Mellin transform:

### Inverse Z Transform(2)

Inverse Z transform of a function:

For :

Obtain the result from its definition as a contour integral:

Inverse Z transform of a function:

For :

From its definition as a contour integral of large radius:

### 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:

It can also be computed with NContourIntegrate:

Numerical contour integrals can also be obtained using NIntegrate:

This is equivalent to:

NIntegrate 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:

Compare with ContourIntegrate:

Another contour integral over a sector of varying radius:

Compare with ContourIntegrate:

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

#### Text

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

#### CMS

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#### APA

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#### BibTeX

@misc{reference.wolfram_2024_ncontourintegrate, author="Wolfram Research", title="{NContourIntegrate}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/NContourIntegrate.html}", note=[Accessed: 23-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_ncontourintegrate, organization={Wolfram Research}, title={NContourIntegrate}, year={2023}, url={https://reference.wolfram.com/language/ref/NContourIntegrate.html}, note=[Accessed: 23-July-2024 ]}