This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# NResidue

 numerically finds the residue of expr near the point .
• The expression expr must be numeric when its argument x is numeric.
• The residue is defined as the coefficient of in the Laurent expansion of expr.
• numerically integrates around a small circle centered at the point in the complex plane. will return an incorrect result when the punctured disk is not analytic.
• 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, can find residues even if the power series cannot be computed.
• 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:
 Out[2]=
 Scope   (2)
can find residues of functions with essential singularities:
Since Series is unable to handle essential singularities, Residue returns unevaluated:
allows for some error in the location of the pole:
Due to machine-precision arithmetic, is not a pole:
With Residue, the error in the location of the pole yields a result of zero: