The asymptotic expansions of Hankel transforms and related integrals
Abstract: In this paper, the asymptotic expansion of integrals of the form s considered, as tends to infinity, and where 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 . The expansion given by Willis for Hankel transforms is seen to be valid under conditions of less restrictive than those imposed by that author.
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-  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
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- H. F. Willis, ``A formula for expanding an integral as a series,'' Philos. Mag., v. (7) 39, 1948, pp. 455-459. MR 10, 22. MR 0025534 (10:22d)
- A. Erdélyi, ``Asymptotic representations of Fourier integrals and the method of stationary phase,'' J. Soc. Indust. Appl. Math., v. 3, 1955, pp. 17-27. MR 17, 29. MR 0070744 (17:29g)
- G. N. Watson, Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, Cambridge; Macmillan, New York, 1944. MR 6, 64. MR 0010746 (6:64a)
- Y. L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York, 1962. MR 25 #5198. MR 0141801 (25:5198)
- 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.
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Keywords: Asymptotic expansion, far-field approximation, integral transforms, Hankel transforms, Bessel functions, cylindrical functions, sine integral, Fresnel integral
Article copyright: © Copyright 1972 American Mathematical Society