Bessel-integral functions

Abstract

§1. Summary. In a very remarkable work on the operational Calculus, Dr Balth. van der Pol 1 has introduced a new function, playing with respect to Bessel function of order zero the same part as the cosine-or sine-integral with respect to the ordinary cosine or sine. He showed that this function-which he called Besselintegral junction-can be used to express the differential coefficient of any Bessel function with respect to its index. But he did not investigate the further properties of his new function. I propose to give here some of them, which appear to be interesting, and to introduce and study the functions connected, in the same way, with Bessel functions of any order. § 2. Definition of the function of order zero.

The Integral Mittag-Leffler, Whittaker and Wright Functions

“…Integrands in the second group of integral special functions include special functions, the most well-known and applied of which are the integral Bessel functions (see, e.g., [3,7,[9][10][11][12][13])…”

On some expansions for bessel integral functions

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Inves tigations by van del' Pol an d Humbert concernin g t he Bessel integral function of order zero ar e extended to Bessel fun ctions of other kinds and to function s r elated to Bessel functions.The B essel integral function of order zero, Various pl'oper ties of this function have been inves tigated by Hum.bed [4] . A number of definite in tegrals, especiall y of the Fourier or Laplace transform type, can be redu ced to (1).

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“…Th e numerical computation of J io(x) requi res the transformation of the integ ral in (1) into an expression involving a series in either ascending powers (fo r not too la rge x) or descending powers (for large x) of the variable ]'. One of these expansions [4,6] is (3) where one can writeAn asymptotic expansion of J io(x) for large x [5 , 7] is listed in (6). The preparation of a set of numerical tables by the Computation Laboratory mad e it necessary to derive expansions of types (3) and (6) for functions defined similarly to (1) but with J o(t) r eplaced by another kind of B essel or related function.…”

“…Various pl'oper ties of this function have been inves tigated by Hum.bed [4] . A number of definite in tegrals, especiall y of the Fourier or Laplace transform type, can be redu ced to (1).…”

“…Th e numerical computation of J io(x) requi res the transformation of the integ ral in (1) into an expression involving a series in either ascending powers (fo r not too la rge x) or descending powers (for large x) of the variable ]'. One of these expansions [4,6] is (3) where one can write…”