Integrals of Bessel functions

Arxiv preprint math/9307213, 1993

A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber's second exponential integral and Laplace transforms of products of three Bessel functions.

Math Comput, 1982

Efficient stratagems are developed for numerically evaluating one-and two-dimensional integrals over x, y with integrand exp(-x-y)I0(2Jpxy). The integrals are expressed in terms of convergent series, which exhibit the correct limiting behavior, and which can be evaluated recursively. The performances of these stratagems are compared with numerical integration.

Applications of Mathematics, 1986

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Communications in Numerical Analysis, 2014

In recent years, several integral formulas involving a variety of special functions have been developed by many authors. Also many integral formulas containing the Bessel function J ν (z) have been presented. Very recently, Rakha et al. presented some generalized integral formulas involving the hypergeometric functions. In this sequel, here, we aim at establishing two generalized integral formulas involving a Bessel functions of the first kind, which are expressed in terms of the generalized Wright hypergeometric function. Some interesting special cases of our main results are also considered.

Communications in Numerical Analysis, 2016

Our focus to presenting two very general integral formulas whose integrands are the integrand given in the Oberhettingers integral formula and a finite product of the generalized Bessel function of the first kind, which are expressed in terms of the generalized Lauricella functions. Among a large number of interesting and potentially useful special cases of our main results, some integral formulas involving such elementary functions are also considered.

Bulletin of the Korean Mathematical Society, 2014

A remarkably large number of integral formulas involving a variety of special functions have been developed by many authors. Also many integral formulas involving various Bessel functions have been presented. Very recently, Choi and Agarwal derived two generalized integral formulas associated with the Bessel function Jν (z) of the first kind, which are expressed in terms of the generalized (Wright) hypergeometric functions. In the present sequel to Choi and Agarwal's work, here, in this paper, we establish two new integral formulas involving the generalized Bessel functions, which are also expressed in terms of the generalized (Wright) hypergeometric functions. Some interesting special cases of our two main results are presented. We also point out that the results presented here, being of general character, are easily reducible to yield many diverse new and known integral formulas involving simpler functions.

2019

In the last decades, various integral formulas associated with Bessel functions of different kinds as well as Bessel functions themselves, have been studied and a noteworthy amount of work can be found in the literature. Following up, we present two definite integral formulas involving the product of generalized Bessel function associated with orthogonal polynomials. Also, some intriguing special cases of our main results have been discussed.

Applied Mathematics and Computation, 2014

In this paper we consider the numerical method for computing the infinite highly oscillatory Bessel integrals of the form R 1 a f ðxÞC v ðxxÞdx, where C v ðxxÞ denotes Bessel function J v ðxxÞ of the first kind, Y v ðxxÞ of the second kind, H ð1Þ v ðxxÞ and H ð2Þ v ðxxÞ of the third kind, f is a smooth function on ½a; 1Þ; lim x!1 f ðkÞ ðxÞ ¼ 0ðk ¼ 0; 1; 2;. . .Þ; x is large and a P 1 x k with k 1. We construct the method based on approximating f by a combination of the shifted Chebyshev polynomial so that the generalized moments can be evaluated efficiently by the truncated formula of Whittaker W function. The method is very efficient in obtaining very high precision approximations if x is sufficiently large. Furthermore, we give the error which depends on the endpoint ''a''. Numerical examples are provided to confirm our results.

Boundary Value Problems, 2013

A remarkably large number of integral formulas involving a variety of special functions have been developed by many authors. Very recently, Ali gave three interesting unified integrals involving the hypergeometric function 2 F 1 . Using Ali's method, in this paper, we present two generalized integral formulas involving the Bessel function of the first kind J ν (z), which are expressed in terms of the generalized (Wright) hypergeometric functions. Some interesting special cases of our main results are also considered. MSC: Primary 33B20; 33C20; secondary 33B15; 33C05

Computers & Mathematics with Applications, 1995

Bessel functions have been generalized in a number of ways and many of these generalizations have been proved to be important tools in applications. In this paper we present a unified treatment, thus proving that many of the seemingly different generalizations may be viewed as particular cases of a two-variable function of the type introduced by Miller during the sixties.