Evaluation of some non-elementary integrals involving the generalized hypergeometric function with some applications

2020/03/16 by Victor Nijimbere, Nijimbere, Victor
Mathematics · #Mathematical functions and polynomials #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations

paper · pdf · doi:10.48550/arxiv.2003.07403

Abstract

The indefinite integral ∫ x^αe^ηxβ _pF_q (a₁, a₂, ⋅⋅⋅ a_p; b₁, b₂, ⋅⋅⋅, b_q; λx^γ)dx, where α, η, β, λ, γ≠0 are real or complex constants and _pF_q is the generalized hypergeometric function, is evaluated in terms of an infinite series involving the generalized hypergeometric function. Related integrals in which the exponential function e^ηxβ is either replaced by the hyperbolic function \cosh(ηx^β) or \sinh(ηx^β), or the sinusoidal function cos(ηx^β) or sin(ηx^β), are also evaluated in terms of infinite series involving the generalized hypergeometric function _pF_q. Some application examples from applied analysis, in which some new Fourier and Laplace integrals (or transforms) are evaluated, are given. The analytical solution of the Orr-Sommerfeld equation (with a linear mean flow background) in the short-wave limit is expressed in terms of some infinite series involving the hypergeometric series ₂F₃. Making use of the hyperbolic and Euler identities, some interesting series identities involving exponential, hyperbolic, trigonometric functions and the generalized hypergeometric function are also derived.

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