Treatment of the ‘pole’ at infinity in classical numerical integration with computer algebra software

Abstract

The error in the classical interpolatory (including Gaussian) numerical integration rules for analytic functions can be considered to be due to a ‘pole’ at infinity. The residue of this ‘pole’ is approximately taken into account by using Taylor-Maclaurin series expansions at the origin- and asymptotic expansions at infinity. The computer algebra system Maple V was found appropriate and used in this task. The cases of the Gauss-Chebyshev and the Gauss-Legendre quadrature rules were considered in some detail. Finally, numerical results in applications both for regular and for Cauchy-type principal value integrals are also presented and the strong improvement in the numerical results by the present approach is illustrated.

Keywords:

• Analytic functions
• asymptotic series expansions
• computer algebra software
• contour integrals
• error term
• expansions
• Gauss-Chebyshev rule
• Gauss-Legendre rule
• numerical integration
• poles
• principal value integrals
• quadrature rules
• Taylor-Maclaurin series expansions

C.R. Categories:

• G.1.4 (primary)
• 1.1.4 (secondary)