Euler Type Integrals and Integrals in Terms of Extended Beta Function

Subuhi Khan,Mustafa Walid Al-Saad

doi:10.1155/2014/686324

Abstract

We derive the evaluations of certain integrals of Euler type involving generalized hypergeometric series. Further, we establish a theorem on extended beta function, which provides evaluation of certain integrals in terms of extended beta function and certain special polynomials. The possibility of extending some of the derived results to multivariable case is also investigated.

Highlights

Euler generalized the factorial function from the domain of natural numbers to the gamma function ∞Γ (α) = ∫ tα−1e−tdt (Re (α) > 0), (1)defined over the right half of the complex plane
The closed-form solutions to a considerable number of problems in applied mathematics, astrophysics, nuclear physics, statistics, and engineering can be expressed in terms of incomplete gamma functions
Chaudhry and Zubair [2] extended the domain of these functions to the entire complex plane by inserting a regularization factor exp(−b/t) in the integrand of (1)

Summary

Introduction

Defined over the right half of the complex plane. This led Legendre (in 1811) to decompose the gamma function into the incomplete gamma functions, γ(α, x) and Γ(α, x) [1], x γ (α, x) = ∫ tα−1e−tdt (Re (α) > 0) ,. The regularizer exp(−b/t) proved very useful in extending the domain of Riemann’s zeta function, thereby providing relationships that could not have been obtained with the original zeta function. In view of the fact that the regularization factor is useful in extending the domain of the gamma and zeta functions. The EBF B(α, β; b) is extremely useful in the sense that most of the properties of the beta function carry over naturally and for it This extension is important due to the fact that this function is related to other special functions for particular values of the variables. For some particular functions f, specially in the symmetric case α = β, are derived in [5] These evaluations are related to various reduction formulae for hypergeometric functions represented by such integrals.

Euler Type Integrals

Integrals in Terms of Extended Beta Function

Concluding Remarks