The asymptotic expansions of Hankel transforms and related integrals
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- by Robert F. MacKinnon PDF
- Math. Comp. 26 (1972), 515-527 Request permission
Abstract:
In this paper, the asymptotic expansion of integrals of the form $\int _0^\infty {F(kr)f(k)dk}$ s considered, as $r$ tends to infinity, and where $F(kr)$ are Bessel functions of the first and second kind, or functions closely related to these. Asymptotic expansions for several functions of this type are presented under suitable restrictions on $f(k)$. The expansion given by Willis for Hankel transforms is seen to be valid under conditions of $f(k)$ less restrictive than those imposed by that author.References
- H. F. Willis, A formula for expanding an integral as a series, Philos. Mag. (7) 39 (1948), 455–459. MR 25534
- A. Erdélyi, Asymptotic representations of Fourier integrals and the method of stationary phase, J. Soc. Indust. Appl. Math. 3 (1955), 17–27. MR 70744
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
- Yudell L. Luke, Integrals of Bessel functions, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1962. MR 0141801 I. S. Gradšteĭn & I. M. Ryžik, Tables of Integrals, Series and Products, 4th ed., Fitzmatgiz, Moscow, 1963; English transl., Academic Press, New York, 1965. MR 28 #5198; MR 33 #5952.
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Math. Comp. 26 (1972), 515-527
- MSC: Primary 44A15
- DOI: https://doi.org/10.1090/S0025-5718-1972-0308695-9
- MathSciNet review: 0308695