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Bessel-integral functions

Published online by Cambridge University Press:  20 January 2009

Pierre Humbert
Pierre Humbert
Affiliation:
University of Montpellier.
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Summary

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In a very remarkable work on the operational Calculus, Dr Balth. van der Pol 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 Bessel-integral 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.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 3 , Issue 4 , August 1933 , pp. 276 - 285
Copyright
Copyright © Edinburgh Mathematical Society 1933

References

page 276 note 1 Philosophical Magazine, 8 (1929), 861–898, (887).Google Scholar

page 276 note 2 For properties of cosine- and sine-integrals, the reader is referred to Nielsen, Niels, Theorie des Integrallgarithmus (Leipzig, 1906). Nielsen's notation will be used through this paper. As regards Bessel functions, we shall follow Watson's notation.Google Scholar

page 276 note 3 Dr van der Pol uses the simple notation Ji (x). As we shall deal with functions connected with the Bessel function of order n, we find it convenient to denote van der Pol's function by Ji 0(x), thus introducing the order of Bessel-integral functions.

page 284 note 1 The third of these formulae is given in Nielsen's book with a slight mistake: he wrote instead of which is the real value.