Abstract
The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, $ \int x^\alpha e^{\eta x^\beta}dx, \int x^\alpha \cosh\left(\eta x^\beta\right)dx, \int x^\alpha \sinh\left(\eta x^\beta\right)dx, \int x^\alpha \cos\left(\eta x^\beta\right)dx$ and $\int x^\alpha \sin\left(\eta x^\beta\right)dx $ where $\alpha, \eta$ and $\beta$ are real or complex constants are evaluated in terms of the confluent hypergeometric function $_1F_1$ and the hypergeometric function $_1F_2$. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions $_1F_1$ and $_1F_2$. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and normal distributions are also obtained. The obtained generalized distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square ($\chi^2$) statistical tests and those based on central limit theorem (CLT)).
Highlights
The confluent hypergeometric function 1 F1 and the hypergeometric function 1 F2 are used throughout this paper
The obtained generalized probability distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square ( χ 2 ) statistical tests and other statistical tests constructed based on the central limit theorem (CLT)), while avoiding the use of computational approximations which are in general expensive and associated with numerical errors
Formulas for non-elementary integrals of the types ∫ xα eηxβ dx, ( ) ( ) ( ) ( ) ∫ xα cosh η xβ dx, ∫ xα sinh η xβ dx, ∫ xα cos η xβ dx and ∫ xα sin η xβ dx where α,η and β are real or complex constants were obtained in terms of the confluent hypergeometric function 1 F1 and the hypergeometric function 1 F2 in section 2
Summary
The confluent hypergeometric function 1 F1 and the hypergeometric function 1 F2 are used throughout this paper. ∫ Daw ( z ) = e−z2 zex dx 0 and other related functions in mathematical physics such Faddeeva, Fried-Conte, Jackson, Fresnel and Gordeyev integrals were analytically evaluated by Nijimbere [7] in terms of the confluent hypergeometric function 1 F1 using the same analytical method as in this study rather than using numerical approximations, see for example [8] [9] Another goal of this work is to obtain some identities (or formulas) involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions 1 F1 and 1 F2 using the Euler and hyperbolic identities. The main results of the paper are given as propositions, theorems and corollaries in Sections 2.1, 2.2, 3.1 and 3.2
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