### Abstract

Asymptotic expansions of certain finite and infinite integrals involving products of two Bessel functions of the first kind are obtained by using the generalized hypergeometric and Meijer functions. The Bessel functions involved are of arbitrary (generally different) orders, but of the same argument containing a parameter which tends to infinity. These types of integrals arise in various contexts, including wave scattering and crystallography, and are of general mathematical interest being related to the Riemann-Liouville and Hankel integrals. The results complete the asymptotic expansions derived previously by two different methods - a straightforward approach and the Mellin-transform technique. These asymptotic expansions supply practical algorithms for computing the integrals. The leading terms explicitly provide valuable analytical insight into the high-frequency behavior of the solutions to the wave-scattering problems.

Original language | English (US) |
---|---|

Pages (from-to) | 533-543 |

Number of pages | 11 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 50 |

Issue number | 1-3 |

DOIs | |

State | Published - May 20 1994 |

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### Keywords

- Asymptotic expansion
- Bessel functions
- Generalized hypergeometric function
- Hankel integral
- Meijer function
- Riemann-Liouville integral

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Numerical Analysis

### Cite this

*Journal of Computational and Applied Mathematics*,

*50*(1-3), 533-543. https://doi.org/10.1016/0377-0427(94)90326-3

**Asymptotic expansions of integrals of two Bessel functions via the generalized hypergeometric and Meijer functions.** / Stoyanov, B. J.; Farrell, R. A.; Bird, J. F.

Research output: Contribution to journal › Article

*Journal of Computational and Applied Mathematics*, vol. 50, no. 1-3, pp. 533-543. https://doi.org/10.1016/0377-0427(94)90326-3

}

TY - JOUR

T1 - Asymptotic expansions of integrals of two Bessel functions via the generalized hypergeometric and Meijer functions

AU - Stoyanov, B. J.

AU - Farrell, R. A.

AU - Bird, J. F.

PY - 1994/5/20

Y1 - 1994/5/20

N2 - Asymptotic expansions of certain finite and infinite integrals involving products of two Bessel functions of the first kind are obtained by using the generalized hypergeometric and Meijer functions. The Bessel functions involved are of arbitrary (generally different) orders, but of the same argument containing a parameter which tends to infinity. These types of integrals arise in various contexts, including wave scattering and crystallography, and are of general mathematical interest being related to the Riemann-Liouville and Hankel integrals. The results complete the asymptotic expansions derived previously by two different methods - a straightforward approach and the Mellin-transform technique. These asymptotic expansions supply practical algorithms for computing the integrals. The leading terms explicitly provide valuable analytical insight into the high-frequency behavior of the solutions to the wave-scattering problems.

AB - Asymptotic expansions of certain finite and infinite integrals involving products of two Bessel functions of the first kind are obtained by using the generalized hypergeometric and Meijer functions. The Bessel functions involved are of arbitrary (generally different) orders, but of the same argument containing a parameter which tends to infinity. These types of integrals arise in various contexts, including wave scattering and crystallography, and are of general mathematical interest being related to the Riemann-Liouville and Hankel integrals. The results complete the asymptotic expansions derived previously by two different methods - a straightforward approach and the Mellin-transform technique. These asymptotic expansions supply practical algorithms for computing the integrals. The leading terms explicitly provide valuable analytical insight into the high-frequency behavior of the solutions to the wave-scattering problems.

KW - Asymptotic expansion

KW - Bessel functions

KW - Generalized hypergeometric function

KW - Hankel integral

KW - Meijer function

KW - Riemann-Liouville integral

UR - http://www.scopus.com/inward/record.url?scp=0028422584&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028422584&partnerID=8YFLogxK

U2 - 10.1016/0377-0427(94)90326-3

DO - 10.1016/0377-0427(94)90326-3

M3 - Article

AN - SCOPUS:0028422584

VL - 50

SP - 533

EP - 543

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1-3

ER -