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

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.</p

A unified presentation of generalised Voigt functions

Voigt functions occur frequently in a wide variety of problems in several diverse fields of physics. This paper presents a unified study of generalised Voigt functions. In particular, some expansions of unified Voigt functions are given in terms of the original functions. Some deductions from these representations are obtained which give us an opportunity to underline the special role of the associated generating functions

Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan Integrals <em>R</em>_<em>S,C</em>(<em>m, n</em>) in Terms of Hypergeometric Series ₂<em>F</em>₃(⋅)

In this chapter, we obtain analytical expressions of infinite Fourier sine and cosine transform-based Ramanujan integrals, RS,Cmn=∫0∞xm−1+exp2πxsincosπnxdx, in an infinite series of hypergeometric functions 2F3⋅, using the hypergeometric technique. Also, we have given some generalizations of the Ramanujan’s integrals RS,Cmn in the form of integrals denoted by IS,C∗υbcλy,JS,Cυbcλy,KS,Cυbcλy and IS,Cυbλy. These generalized definite integrals are expressed in terms of ordinary hypergeometric functions 2F3⋅, with suitable convergence conditions. Moreover, as applications of Ramanujan’s integrals RS,Cmn, some closed form of infinite summation formulas involving hypergeometric functions 1F2, 2F3⋅, and 0F1 are derived