Numerical Calculus Package Symbol

NResidue

NResidue[expr, {z, z₀}]
numerically finds the residue of expr near the point z=z₀.

To use NResidue, you first need to load the Numerical Calculus Package using Needs["NumericalCalculus`"].

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

Residue of the function

about the origin:

Needs["NumericalCalculus`"]

Residue of the function

about the origin:

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:

NResidue threads element-wise over lists:

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:

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

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

Residues of numerical functions:

NSeries can also compute residues of numerical functions:

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