This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)


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: