Abstract
Integration by parts is one of the most popular techniques in the analysis of integrals and is one of the simplest methods to generate asymptotic expansions of integral representations. The product of the technique is usually a divergent series formed from evaluating boundary terms; however, sometimes the remaining integral is also evaluated. Due to the successive differentiation and anti-differentiation required to form the series or the remaining integral, the technique is difficult to apply to problems more complicated than the simplest. In this contribution, we explore a generalized and formalized integration by parts to create equivalent representations to some challenging integrals.As a demonstrative archetype, we examine Bessel integrals, Fresnel integrals and Airy functions.
Highlights
It is well known that in applied mathematics and in the numerical treatment of scientific problems, slowly convergent integrals occur abundantly
The popular candidates were the summation of the generally divergent boundary terms, the transformed integral has been used in cases where the boundary terms vanished
The selection of an appropriate sequence {xl}nl=0 for the integration subintervals in the Staircase algorithm is critical. This sequence should leave the integral subintervals to be numerically integrable – the points cannot be spaced so far apart that a quadrature routine is incapable of providing the solution
Summary
It is well known that in applied mathematics and in the numerical treatment of scientific problems, slowly convergent integrals occur abundantly They are produced by approximation procedures depending on a parameter, iterative methods, quadrature schemes, perturbation techniques and reliable evaluation of functions defined by integrals. Is often discarded and the divergent series has either been summed straightforwardly or summed through the use of sequence transformations Another example of the use of integration by parts in numerical integration arises in [19], applied to the oscillatory spherical Bessel integral function involved in molecular integrals. For the computation of the integral representations, we propose a robust algorithm which will be referred to as the staircase algorithm The algorithm uses both the boundary terms and the transformed integrals and progressively descends the integrand to an asymptotically favourable situation by applying the reformalized integration by parts at each iteration.
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