NumericalCalculus`
NumericalCalculus`

# NResidue

NResidue[expr,{z,z0}]

numerically finds the residue of expr near the point z=z0.

# Details and Options

• To use NResidue, you first need to load the Numerical Calculus Package using Needs["NumericalCalculus`"].
• The expression expr must be numeric when its argument x is numeric.
• The residue is defined as the coefficient of (z-z0)-1 in the Laurent expansion of expr.
• NResidue numerically integrates around a small circle centered at the point z0 in the complex plane. NResidue will return an incorrect result when the punctured disk is not analytic.
• NResidue is unable to recognize small numbers that should in fact be zero. Chop is often needed to eliminate these spurious residuals.
• Although Residue usually needs to be able to evaluate power series at a point, NResidue can find residues even if the power series cannot be computed.
• NResidue has the same options as NIntegrate, with the following additions and changes:
•  Radius 1/100 radius of contour on which integral is evaluated Method Trapezoidal integration method to use

# Examples

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## Basic Examples(1)

Residue of the function about the origin:

## Scope(2)

NResidue can find residues of functions with essential singularities:

Since Series is unable to handle essential singularities, Residue returns unevaluated:

NResidue allows for some error in the location of the pole:

Due to machine-precision arithmetic, z -> 1. is not a pole:

With Residue, the error in the location of the pole yields a result of zero:

## Options(3)

Use Radius to shrink the radius of the contour of integration to isolate a single pole:

Increase the radius to improve convergence of the integration if no other poles are nearby:

### WorkingPrecision(1)

NResidue accepts options for NIntegrate, which are sometimes necessary to get an accurate result:

## Applications(2)

Use NResidue to evaluate derivatives of a function evaluated at a point:

Residues of numerical functions:

## Properties & Relations(1)

NSeries can also compute residues of numerical functions:

Using NSeries:

## Possible Issues(1)

NResidue will return an incorrect result when the integration contour contains branch cuts:

Wolfram Research (2007), NResidue, Wolfram Language function, https://reference.wolfram.com/language/NumericalCalculus/ref/NResidue.html.

#### Text

Wolfram Research (2007), NResidue, Wolfram Language function, https://reference.wolfram.com/language/NumericalCalculus/ref/NResidue.html.

#### CMS

Wolfram Language. 2007. "NResidue." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/NumericalCalculus/ref/NResidue.html.

#### APA

Wolfram Language. (2007). NResidue. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/NumericalCalculus/ref/NResidue.html

#### BibTeX

@misc{reference.wolfram_2022_nresidue, author="Wolfram Research", title="{NResidue}", year="2007", howpublished="\url{https://reference.wolfram.com/language/NumericalCalculus/ref/NResidue.html}", note=[Accessed: 28-March-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_nresidue, organization={Wolfram Research}, title={NResidue}, year={2007}, url={https://reference.wolfram.com/language/NumericalCalculus/ref/NResidue.html}, note=[Accessed: 28-March-2023 ]}