EVALUATION OF THE NON-ELEMENTARY INTEGRAL ∫eλx^α, α≥2, AND OTHER RELATED INTEGRALS

Abstract

A formula for the non-elementary integral e λx α dx where α is real and greater or equal two, is obtained in terms of the confluent hypergeometric function 1 F 1 by expanding the integrand as a Taylor series. This result is verified by directly evaluating the area under the Gaussian Bell curve, corresponding to α = 2, using the asymptotic expression for the confluent hypergeometric function and the Fundamental Theorem of Calculus (FTC). Two different but equivalent expressions, one in terms of the confluent hypergeometric function 1 F 1 and another one in terms of the hypergeometric function 1 F 2 , are obtained for each of these integrals, cosh(λx α )dx, sinh(λx α )dx, cos(λx α )dx and sin(λx α )dx, λ ∈ C, α ≥ 2. And the hypergeometric function 1 F 2 is expressed in terms of the confluent hypergeometric function 1 F 1 . Some of the applications of the non-elementary integral e λx α dx, α ≥ 2 such as the Gaussian distribution and the Maxwell-Bortsman distribution are given.

Analytical and asymptotic evaluations of Dawson’s integral and related functions in mathematical physics

“…We use Theorem 1 in Nijimbere [20] to express (2) in terms of the confluent hypergeometric functions 1 F 1 , and then obtain…”

“…First, it is important to point out that Dawson's integral and Faddeeva's integral are non-elementary integrals. Being non-elementary means that they cannot neither be expressed in terms of elementary functions such polynomials of finite degree, exponentials and logarithms, nor in terms of mathematical expressions obtained by performing finite algebraic combinations involving elementary functions [16,20,23]. For this reason, it is not possible to evaluate analytically these integrals in closed form or, in other words, in terms of elementary functions [3,16,20,23].…”

“…Being non-elementary means that they cannot neither be expressed in terms of elementary functions such polynomials of finite degree, exponentials and logarithms, nor in terms of mathematical expressions obtained by performing finite algebraic combinations involving elementary functions [16,20,23]. For this reason, it is not possible to evaluate analytically these integrals in closed form or, in other words, in terms of elementary functions [3,16,20,23]. To this end, intensive works have mainly focused on numerical approximations [2,3,7,8,9,10,12,17,15,22,27,28].…”

“…None of the above integrals can be evaluated analytically in close form, or in terms of elementary functions, as pointed out by Abrarov and Quine [3] of course, but one can express them in terms of a special function, the confluent hypergeometric function 1 F 1 [1,19]. Noting that Nijimbere [20] (Theorem 1) has evaluated the non-elementary integral b a e λx α dx, α ≥ 2 for any constant λ in terms of the confluent hypergeometric function 1 F 1 , as a first objective of this paper, we use the results in Nijimbere [20] to obtain an analytical expression for Dawson's integral in terms of the confluent hypergeometric function 1 F 1 . And hence, we write Faddeeva's integral and the above related integrals in terms of 1 F 1 .…”

Evaluation of Some Non-Elementary Integrals Involving Sine, Cosine, Exponential and Logarithmic Integrals: Part I

“…To this end, in this paper, formulas for these non-elementary integrals are expressed in terms of the hypergeometric functions 1 F 2 and 2 F 3 whose properties, for example, the asymptotic expansions for large argument (|λx| ≫ 1), are known [9]. We do so by expanding the integrand in terms of its Taylor series and by integrating the series term by term as in [7]. And therefore, their corresponding definite integrals can be evaluated using the Fundamental Theorem of Calculus (FTC).…”