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

TL;DR

This paper evaluates complex integrals involving the generalized hypergeometric function and related functions, providing new series solutions and applications in Fourier, Laplace transforms, and fluid stability analysis.

Contribution

It introduces novel series representations for integrals with hypergeometric functions and applies them to solve the Orr-Sommerfeld equation and derive new series identities.

Findings

Series expressions for integrals involving hypergeometric functions.

Application to Fourier and Laplace transforms in analysis.

Analytical solutions for the Orr-Sommerfeld equation.

Abstract

The indefinite integral $$ \int x^\alpha e^{\eta x^\beta}\,_pF_q (a_1, a_2, \cdot\cdot\cdot a_p; b_1, b_2, \cdot\cdot\cdot, b_q; \lambda x^{\gamma})dx, $$ where $\alpha, \eta, \beta, \lambda, \gamma\ne0$ 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 $e^{\eta x^\beta}$ is either replaced by the hyperbolic function $\cosh\left(\eta x^\beta\right)$ or $\sinh\left(\eta x^\beta\right)$, or the sinusoidal function $\cos\left(\eta x^\beta\right)$ or $\sin\left(\eta x^\beta\right)$, 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 $_2F_3$. 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.