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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Computation of integrals with oscillatory and singular integrands
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by Bing Yuan Ting and Yudell L. Luke PDF
Math. Comp. 37 (1981), 169-183 Request permission

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 $G(\theta ) = \smallint _a^bw(x){e^{i\theta x}}f(x) dx$, where a and b can be finite or infinite, $\theta$ is a parameter which is usually large, $f(x)$ is analytic in the range of integration, and the singularities are encompassed in the weight function $w(x)$. We suppose that $f(x)$ can be expanded in series of polynomials which are orthogonal over the interval of integration with respect to $w(x)$. There are two such expansions for $f(x)$. 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.
References
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Additional Information
  • © Copyright 1981 American Mathematical Society
  • 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