Abstract
In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s ζ(3) constants are produced. Some special cases of the result in terms of fundamental constants are evaluated. All the results in this work are new.
Highlights
Bessel functions were first studied by Daniel Bernoulli [1] and generalized by Friedrich Bessel [2] and are canonical solutions of Bessel’s differential equation (see section (10.13) in [3])
Bessel functions arise in the application of cylindrical symmetry in which the physics is described by Laplace’s equation (see section (10.73) in [3])
We have presented a novel method for deriving a new quadruple integral involving the product of Bessel functions along with some interesting special cases using contour integration
Summary
Bessel functions were first studied by Daniel Bernoulli [1] and generalized by Friedrich Bessel [2] and are canonical solutions of Bessel’s differential equation (see section (10.13) in [3]). Bessel functions arise in the application of cylindrical symmetry in which the physics is described by Laplace’s equation (see section (10.73) in [3]). The definite integral of the product of Bessel functions, which find importance in many branches of mathematical physics, elasticity, potential theory and applied probability, is studied in the works of Glasser [4] and Chaudhry et al [5]. Our goal is to expand upon the current literature of multiple integrals involving the product of Bessel functions by providing a formal derivation in terms of the. Any new result on multiple integrals of the product of Bessel functions is important because of their many applications in applied and pure mathematics
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