Computation of integrals with oscillatory and singular integrands

Authors:
Bing Yuan Ting and Yudell L. Luke

Journal:
Math. Comp. **37** (1981), 169-183

MSC:
Primary 65D30; Secondary 41A60

DOI:
https://doi.org/10.1090/S0025-5718-1981-0616369-5

MathSciNet review:
616369

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Abstract: This paper is concerned with evaluation of integrals whose integrands are oscillatory and contain singularities at the endpoints of the interval of integration. A typical form is , where *a* and *b* can be finite or infinite, is a parameter which is usually large, is analytic in the range of integration, and the singularities are encompassed in the weight function . We suppose that can be expanded in series of polynomials which are orthogonal over the interval of integration with respect to . There are two such expansions for . One is an infinite series which follows from the usual orthogonality property. The other is a polynomial approximation plus a remainder. The relations between the coefficients in these representations are detailed and methods for the evaluation of these are analyzed. Error analyses are provided. A numerical example is given to illustrate the effectiveness of the schemes developed.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1981-0616369-5

Keywords:
Fourier coefficients,
numerical quadrature,
integration of oscillatory and singular integrands,
interpolation

Article copyright:
© Copyright 1981
American Mathematical Society