This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
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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:
Radius1/100radius of contour on which integral is evaluated
MethodTrapezoidalintegration method to use
Residue of the function about the origin:
Residue of the function about the origin:
Click for copyable input
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:
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:
accepts options for NIntegrate, which are sometimes necessary to get an accurate result:
Use to evaluate derivatives of a function evaluated at a point:
Residues of numerical functions:
NSeries can also compute residues of numerical functions:
Using NSeries:
will return an incorrect result when the integration contour contains branch cuts: