A treatise on the theory of Bessel functions, by G. N. Watson.

CHAPTER XIII

INFINITE INTEGRALS
13'1. Various types of infinite integrals.
The subject of this chapter is the investigation of various classes of infinite
integrals which contain either Bessel functions or functions of a similar character
under the integral sign. The methods of evaluating such integrals are not
very numerous; they consist, for the most part, of the following devices:
(I) Expanding the Bessel function in powers of its argument and integrating term-by-term.
(II) Replacing the Bessel function by Poisson's integral, changing the order
of the integrations, and then carrying out the integrations.
(III) Replacing the Bessel function by one of the generalisations of Bessel's
integral, changing the order of the integrations, and then carrying out the
integrations; this procedure has been carried out systematically by Sonine*
in his weighty memoir.
(IV) When two Bessel functions of the same order occur as a product
under the integral sign, they may be replaced by the integral of a single
Bessel function by Gegenbauer's formula (cf. ~ 12'1), and the order of the integrations is then changedt.
(V) When two functions of different orders but of the same argument
occur as a product under the integral sign, the product may be replaced by
the integral of a single Bessel function by Neumann's formula (~ 5'43), and
the order of the integrations is then changed.
(VI) The Bessel function under the integral sign may be replaced by the
contour integral of Barnes' type (~ 6'5) involving Gamma functions, and the
order of the integrations is then changed; this very powerful method has not
previously been investigated in a systematic manner.
Infinite integrals involving Bessel functions under the integral sign are
not only of great interest to the Pure Mathematician, but they are of extreme
importance in many branches of Mathematical Physics. And the various types
are so numerous that it is not possible to give more than a selection of the
most important integrals, whose values will be worked out by the most suitable
methods; care has been taken to evaluate several examples by each method.
In spite of the incompleteness of this chapter, its length must be contrasted
unfavourably with the length of the chapter on finite integrals.
Math. Ann. xvi. (1880), pp. 33-60.
f This procedure has been carried out by Gegenbauer in a number of papers published in the
Wiener Sitzungsberichte.