On finite integrals involving trigonometric, Bessel, and Legendre functions
HTML articles powered by AMS MathViewer
- by Richard L. Lewis PDF
- Math. Comp. 23 (1969), 259-273 Request permission
Abstract:
A finite integral involving the product of powers of trigonometric functions, up to two associated Legendre functions, and zero or one Bessel function is evaluated. When certain combinations of the otherwise complex function parameters are integers, the resulting expression becomes greatly simplified. So restricting the parameters, this still quite general case may be transformed into four canonical forms, each of which admits rapid convergence of the only nonterminating series in the expressions. Finally, closed form expressions are obtained for a number of special cases.References
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756 B. W. Barnes, “On generalized Legendre functions,” Quart. J. Math. Oxford Ser., v. 31, 1908, pp. 97–204. B. I. Whittaker & G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, New York, 1927. A. L. Dixon & W. L. Ferrar, Quart. J. Math. Oxford Ser., v. 7, 1936, pp. 81–96.
- 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 W. N. Bailey, Generalized Hypergeometric Series, Cambridge Univ. Press, New York, 1935. F. J. W. Whipple, “A group of generalized hypergeometric series,” Proc. London Math. Soc. (2), v. 23, 1925, pp. 104–114. L. Saalschütz, Zeitschrift für Math, und Phys., v. 35, 1890, pp. 186–188.
- T. M. MacRobert, Formulae for generalized hypergeometric functions as particular cases of more general formulae, Philos. Mag. 28 (1939), 488–492. MR 0000707
- Moshe Mangad, Some limiting values and two error estimation procedures for power series approximations, Math. Comp. 21 (1967), 423–430. MR 224751, DOI 10.1090/S0025-5718-1967-0224751-6
- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0065685 A. Gaunt, “The triplets of helium,” Philos. Trans. Roy. Soc. Ser. A, v. 228, 1929, pp. 192–196.
- D. B. Sears, On the transformation theory of hypergeometric functions and cognate trigonometrical series, Proc. London Math. Soc. (2) 53 (1951), 138–157. MR 41980, DOI 10.1112/plms/s2-53.2.138
Additional Information
- © Copyright 1969 American Mathematical Society
- Journal: Math. Comp. 23 (1969), 259-273
- MSC: Primary 65.25
- DOI: https://doi.org/10.1090/S0025-5718-1969-0242350-9
- MathSciNet review: 0242350