A quadruple integral involving the product of generalized parabolic cylinder functions $D_{v}(\beta x)D_{u}(\alpha z)$: Derivation and evaluation
HTML articles powered by AMS MathViewer
- by Robert Reynolds and Allan Stauffer HTML | PDF
- Proc. Amer. Math. Soc. Ser. B 9 (2022), 174-179
Abstract:
The aim of the present document is to evaluate a quadruple integral involving the product of the generalized Parabolic Cylinder functions $D_{v}(\beta x)D_{u}(\alpha z)$ expressed in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. All the results in this work are new.References
- R. Reynolds, and A. Stauffer, A method for evaluating definite integrals in terms of special functions with examples, Int. Math. Forum 15 (2020), 235β244, DOI:10.12988/imf.2020.91272
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed., Academic Press, Inc., San Diego, CA, 2000. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. MR 1773820
- Keith Oldham, Jan Myland, and Jerome Spanier, An atlas of functions, 2nd ed., Springer, New York, 2009. With Equator, the atlas function calculator; With 1 CD-ROM (Windows). MR 2466333, DOI 10.1007/978-0-387-48807-3
- Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
- Yu. A. Brychkov, O. I. Marichev, and N. V. Savischenko, Handbook of Mellin transforms, Advances in Applied Mathematics, CRC Press, Boca Raton, FL, 2019. MR 3890103
- Herbert Buchholz, The confluent hypergeometric function with special emphasis on its applications, Springer Tracts in Natural Philosophy, Vol. 15, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by H. Lichtblau and K. Wetzel. MR 0240343, DOI 10.1007/978-3-642-88396-5
- G. E. Barr, A note on integrals involving parabolic cylinder functions, SIAM J. Appl. Math. 16 (1968), 71β74. MR 222355, DOI 10.1137/0116005
- B. D. Sleeman, On parabolic cylinder functions, J. Inst. Math. Appl. 4 (1968), 106β112. MR 227486, DOI 10.1093/imamat/4.1.106
- Cyril Malyshev, A Nicholson-type integral for the product of two parabolic cylinder functions $D_\nu (x)D_\nu (-x)$ at $\Re \nu <0$, Integral Transforms Spec. Funct. 14 (2003), no.Β 2, 139β148. MR 1969841, DOI 10.1080/1065246031000074371
Additional Information
- Robert Reynolds
- Affiliation: Department of Mathematics and Statistics, York University, Toronto, Ontario M3J1P3, Canada
- MR Author ID: 1404777
- ORCID: 0000-0002-4230-9925
- Email: milver@my.yorku.ca
- Allan Stauffer
- Affiliation: Department of Mathematics and Statistics, York University, Toronto, Ontario M3J1P3, Canada
- MR Author ID: 222395
- Email: stauffer@yorku.ca
- Received by editor(s): December 28, 2021
- Received by editor(s) in revised form: February 9, 2022
- Published electronically: April 18, 2022
- Additional Notes: This research was supported by NSERC Canada under Grant 504070
- Communicated by: Mourad Ismail
- © Copyright 2022 by the authors under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License (CC BY NC ND 4.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 9 (2022), 174-179
- MSC (2020): Primary 30E20, 33-01, 33-03, 33-04, 33E20
- DOI: https://doi.org/10.1090/bproc/126
- MathSciNet review: 4409299