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 1/100radius of contour on which integral is evaluated
    MethodTrapezoidalintegration method to use


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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:

Generalizations & Extensions  (1)

NResidue threads element-wise over lists:

Options  (3)

Radius  (2)

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,


Wolfram Research (2007), NResidue, Wolfram Language function,


Wolfram Language. 2007. "NResidue." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2007). NResidue. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2022_nresidue, author="Wolfram Research", title="{NResidue}", year="2007", howpublished="\url{}", note=[Accessed: 28-March-2023 ]}


@online{reference.wolfram_2022_nresidue, organization={Wolfram Research}, title={NResidue}, year={2007}, url={}, note=[Accessed: 28-March-2023 ]}