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On a certain class of bessel integrals

In this paper, we attempt to have a clear way of proving some of these results . In fact, we consider a certain class of Bessel integrals where we prove that such integrals vanish under certain conditions. | Turk J Math 28 (2004) , 399 – 413. ¨ ITAK ˙ c TUB On A Certain Class of Bessel Integrals Ali A. Al-Jarrah∗ , A. Al-Momani Abstract There are many old results of integrals involving Bessel functions, currently available in handbooks, but we found no recourse in the well-known references to how they were established. In this paper, we attempt to have a clear way of proving some of these results . In fact, we consider a certain class of Bessel integrals where we prove that such integrals vanish under certain conditions. To this end some theorems regarding this class of integrals with their proofs are put forward. A computer algorithm is provided to implement some of our results. The result in this paper extend the work in [3], and it concludes by indicating the wide range of old and new results that can be obtained. Key Words: Bessel Functions, Infinite Integrals, Integral Representations. The authors wish to thank the referee for his valuable remarks. 1. Introduction and Statement of Results. Bessel functions, with their manifold applications, have been studied in great detail, and extensive tables of these functions are available [2, 5, 6, 8, 9]. Infinite integrals of these functions frequently occur in the investigation of some physical and Engineering problems. There exists a considerable body of information on the subject of these integrals. Of special significance are chapter XIII of Watson’s classical treaties [ 9] and the excellent book by Luke [6]which provides a thorough summary of results prior to 1962. AMS 2000 Subject Classification: 33C10 author. ∗ Corresponding 399 AL-JARRAH, AL-MOMANI The purpose of this paper is to derive a number of old and new infinite integrals involving Bessel functions. In fact, we investigate the class of integrals: Z∞ tµ+1 Jv (at) F (t) dt, (t2 + b2 )m (1) 0 where Jv (at) is the Bessel function of the first kind, and F (t) is a function that has a meromorphic extension in the right upper half-plane. In [3], a .