Movable singularities and quadrature
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- by R. F. Goodrich and F. Stenger PDF
- Math. Comp. 24 (1970), 283-300 Request permission
Abstract:
A general procedure is described for treating a movable singularity in an integral. This enabĺes us to change the original integral ${I_0}$ into $G{I_1}$, where $G$ depends only on the parameters of the singularity and ${I_1}$, is a new integral which exists for all values of the parameters. The results are then applied to the particular problem of evaluating \[ \int _{ - 1}^1 {\frac {{f(x)dx}} {{{{\{ (1 - {x^2})(1 - {k^2}{x^2})\} }^{1/2}}}}} ,\] where $f$ is entire and $k$ varies between $0$ and $1$. Some new quadrature schemes and new effective methods of evaluating incomplete elliptic integrals are derived.References
- Philip J. Davis and Philip Rabinowitz, Numerical integration, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1967. MR 0211604
- H. Hönl, A. W. Maue, and K. Westpfahl, Theorie der Beugung, Handbuch der Physik, Band XXV/1, Springer, Berlin, 1961 (German). MR 0177615
- Frank Stenger, Bounds on the error of Gauss-type quadratures, Numer. Math. 8 (1966), 150–160. MR 196936, DOI 10.1007/BF02163184 C. Caratheodory, Conformal Representation, 2nd ed., Cambridge Tracts in Math. and Math. Physics, no. 28, Cambridge Univ. Press, New York, 1958. MR 13 #734. W. Gautschi, Algorithm $\ldots$ Gaussian Quadrature Formulas (Submitted for publication.)
- John McNamee, Error-bounds for the evaluation of integrals by the Euler-Maclaurin formula and by Gauss-type formulae, Math. Comp. 18 (1964), 368–381. MR 185804, DOI 10.1090/S0025-5718-1964-0185804-1
- Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and physicists, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Band LXVII, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954. MR 0060642
- H. Kober, Dictionary of conformal representations, Dover Publications, Inc., New York, N.Y., 1952. MR 0049326 E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Clarendon Press, Oxford, 1950.
- Wyman G. Fair and Yudell L. Luke, Rational approximations to the incomplete elliptic integrals of the first and second kinds, Math. Comp. 21 (1967), 418–422. MR 222348, DOI 10.1090/S0025-5718-1967-0222348-5
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
- L. Fox, Romberg integration for a class of singular integrands, Comput. J. 10 (1967), 87–93. MR 213013, DOI 10.1093/comjnl/10.1.87
- Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0201039
- Yudell L. Luke, Approximations for elliptic integrals, Math. Comp. 22 (1968), 627–634. MR 226825, DOI 10.1090/S0025-5718-1968-0226825-3
- Pierre Barrucand, Quadratures numériques, fonctions elliptiques et facteur de convergence, C. R. Acad. Sci. Paris 258 (1964), 2742–2744 (French). MR 168117
- F. W. J. Olver, Error bounds for the Liouville-Green (or $WK\,B$) approximation, Proc. Cambridge Philos. Soc. 57 (1961), 790–810. MR 130452
- Günter Meinardus, Approximation von Funktionen und ihre numerische Behandlung, Springer Tracts in Natural Philosophy, Vol. 4, Springer-Verlag, Berlin-New York, 1964 (German). MR 0176272
- A. F. Timan, Teorij pribli+enij funkciĭ deĭstvitel’nogo peremennogo, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960 (Russian). MR 0117478
- R. E. Barnhill and J. A. Wixom, Quadratures with remainders of minimum norm. I, Math. Comp. 21 (1967), 66–75. MR 223089, DOI 10.1090/S0025-5718-1967-0223089-0
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Math. Comp. 24 (1970), 283-300
- MSC: Primary 65.55
- DOI: https://doi.org/10.1090/S0025-5718-1970-0275669-4
- MathSciNet review: 0275669