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SIGMA 8 (2012), 077, 10 pages arXiv:1210.0041
https://doi.org/10.3842/SIGMA.2012.077
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”
Definite Integrals using Orthogonality and Integral Transforms
Howard S. Cohl a and Hans Volkmer b
a) Applied and Computational Mathematics Division, Information Technology Laboratory, National Institute of Standards and Technology,
Gaithersburg, MD, 20899-8910, USA
b) Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI, 53201, USA
Received July 31, 2012, in final form October 15, 2012; Published online October 19, 2012
Abstract
We obtain definite integrals for products of associated Legendre functions
with Bessel functions, associated Legendre functions, and Chebyshev polynomials
of the first kind using orthogonality and integral transforms.
Key words:
definite integrals; associated Legendre functions; Bessel functions; Chebyshev polynomials of the first kind.
pdf (328 kb)
tex (14 kb)
References
- Abramowitz M., Stegun I.A., Handbook of mathematical functions with formulas,
graphs, and mathematical tables, National Bureau of Standards Applied
Mathematics Series, Vol. 55, U.S. Government Printing Office, Washington,
D.C., 1964.
- Askey R., Orthogonal polynomials and special functions, Society for Industrial
and Applied Mathematics, Philadelphia, Pa., 1975.
- Cohl H.S., Fourier, Gegenbauer and Jacobi expansions for a power-law
fundamental solution of the polyharmonic equation and polyspherical addition
theorems, arXiv:1209.6047.
- Cohl H.S., Erratum: Developments in determining the gravitational potential
using toroidal functions, Astronom. Nachr. 333 (2012),
784-785.
- Cohl H.S., Dominici D.E., Generalized Heine's identity for complex Fourier
series of binomials, Proc. R. Soc. Lond. Ser. A 467 (2011),
333-345, arXiv:0912.0126.
- Cohl H.S., Tohline J.E., Rau A.R.P., Srivastava H.M., Developments in
determining the gravitational potential using toroidal functions,
Astronom. Nachr. 321 (2000), 363-372.
- Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, seventh
ed., Elsevier/Academic Press, Amsterdam, 2007.
- Hardy G.H., Further researches in the theory of divergent series and integrals,
Trans. Cambridge Philos. Soc. 21 (1908), 1-48.
- MacRobert T.M., Spherical harmonics. An elementary treatise on harmonic
functions with applications, 2nd ed., Methuen & Co. Ltd., London, 1947.
- Magnus W., Oberhettinger F., Soni R.P., Formulas and theorems for the special
functions of mathematical physics, 3rd ed., Die Grundlehren der
mathematischen Wissenschaften, Bd. 52, Springer-Verlag, New York, 1966.
- Morse P.M., Feshbach H., Methods of theoretical physics, Vols. 1, 2,
McGraw-Hill Book Co. Inc., New York, 1953.
- Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook
of mathematical functions, Cambridge University Press, Cambridge, 2010.
- Prudnikov A.P., Brychkov Y.A., Marichev O.I., Integrals and series.
Vol. 3. More special functions, Gordon and Breach Science Publishers, New
York, 1990.
- Schäfke F.W., Einführung in die Theorie der speziellen Funktionen der
mathematischen Physik, Die Grundlehren der mathematischen Wissenschaften,
Bd. 118, Springer-Verlag, Berlin, 1963.
- Watson G.N., A treatise on the theory of Bessel functions, 2nd ed., Cambridge
Mathematical Library, Cambridge University Press, Cambridge, 1944.
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