A note on a generalisation of a definite integral involving the Bessel function of the first kind

Showkat Ahmad Dar; M. Kamarujjama; R. B. Paris; M. A. Khanday

doi:10.7153/jca-2024-24-10

A note on a generalisation of a definite integral involving the Bessel function of the first kind

Showkat Ahmad Dar^*, M. Kamarujjama, R. B. Paris, M. A. Khanday

^*Corresponding author for this work

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Abstract

We consider a generalisation of a definite integral involving the Bessel function of the first kind. It is shown that this integral can be expressed in terms of the Fox-Wright function _pΨ_q(z) of one variable. Some consequences of this representation are explored by suitable choice of parameters. Further, we compte the range of numerical approximation values of the Ramanujan’s cosine integral _φC (m,n) and sine integral _φS (m,n) for distinct values of m and n by Wolfram Mathematica software. In addition, two closed-form evaluations of infinite series of the Fox-Wright function are deduced and these sums have been verified numerically using Mathematica.

Original language	English
Article number	24-10
Pages (from-to)	179-188
Number of pages	10
Journal	Journal of Classical Analysis
Volume	24
Issue number	2
Early online date	1 Apr 2024
DOIs	https://doi.org/10.7153/jca-2024-24-10
Publication status	Published - 1 Apr 2024

Keywords

Bessel function
Fox-Wright function
Hypergeometric function
Fourier sine and cosine transform
Mellin transform
Computational aspects of special functions

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Cite this

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abstract = "We consider a generalisation of a definite integral involving the Bessel function of the first kind. It is shown that this integral can be expressed in terms of the Fox-Wright function pΨq(z) of one variable. Some consequences of this representation are explored by suitable choice of parameters. Further, we compte the range of numerical approximation values of the Ramanujan{\textquoteright}s cosine integral φC (m,n) and sine integral φS (m,n) for distinct values of m and n by Wolfram Mathematica software. In addition, two closed-form evaluations of infinite series of the Fox-Wright function are deduced and these sums have been verified numerically using Mathematica.",

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AU - Khanday, M. A.

N1 - © 2024, Ele-Math. All rights reserved. Copyright on any research article in a journal published by an Element Open Access journal is retained by the author(s). This journal article is published under the terms of the Creative Commons Attribution Non Commercial License which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes. Data availability statement: Not present.

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N2 - We consider a generalisation of a definite integral involving the Bessel function of the first kind. It is shown that this integral can be expressed in terms of the Fox-Wright function pΨq(z) of one variable. Some consequences of this representation are explored by suitable choice of parameters. Further, we compte the range of numerical approximation values of the Ramanujan’s cosine integral φC (m,n) and sine integral φS (m,n) for distinct values of m and n by Wolfram Mathematica software. In addition, two closed-form evaluations of infinite series of the Fox-Wright function are deduced and these sums have been verified numerically using Mathematica.

AB - We consider a generalisation of a definite integral involving the Bessel function of the first kind. It is shown that this integral can be expressed in terms of the Fox-Wright function pΨq(z) of one variable. Some consequences of this representation are explored by suitable choice of parameters. Further, we compte the range of numerical approximation values of the Ramanujan’s cosine integral φC (m,n) and sine integral φS (m,n) for distinct values of m and n by Wolfram Mathematica software. In addition, two closed-form evaluations of infinite series of the Fox-Wright function are deduced and these sums have been verified numerically using Mathematica.

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A note on a generalisation of a definite integral involving the Bessel function of the first kind

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