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

CHAPTER VI

INTEGRAL REPRESENTATIONS OF BESSEL FUNCTIONS
6'1. Generalisations of Poisson's integral.
In this chapter we shall study various contour integrals associated with
Poisson's integral (~ 2'3, 33) and Bessel's integral (~ 2'2). By suitable choices
of the contour of integration, large numbers of elegant formulae can be obtained
which express Bessel functions as definite integrals. The contour integrals will
also be applied in Chapters VII and viII to obtain approximate formulae and
asymptotic expansions for J, (z) when z or v is large.
It happens that the applications of Poisson's integral are of a more
elementary character than the applications of Bessel's integral, and accordingly
we shall now study integrals of Poisson's type, deferring the study of integrals
of Bessel's type to ~ 6'2. The investigation of generalisations of Poisson's
integral which we shall now give is due in substance to Hankel*.
The simplest of the formulae of ~ 3'3 is ~ 3'3 (4), since this formula contains
a single exponential under the integral sign, while the other formulae contain
circular functions, which are expressible in terms of two exponentials. We
shall therefore examine the circumstances in which contour integrals of the type
z eizt Tdt
are solutions of Bessel's equation; it is supposed that T is a function of t but
not of z, and that the end-points, a and b, are complex numbers independent
of z.
The result of operating on the integral with Bessel's differential operator
V,, defined in ~ 3'1, is as follows:, {zvj eit Tdt} = z+2 eit (1- t2) dt + (2v + 1) izv+l ei Ttdt
C J/ ta Cd aC
= i^+i [eLt T (t2 - 1)j + iv+l i eit (2v + 1) Tt - T(t2 - 1)} dt
* Math. Ann. I. (1869), pp. 473-485. The discussion of the corresponding integrals for Iv (z)
and Kv (z) is due to Schlafli, Ann. di Mat. (2) I. (1868), pp. 232-242, though Schlbfli's results
are expressed in the notation explained in ~ 4-15. The integrals have also been examined in great
detail by Gubler, Zurich Vierteljahrsschrift, xxxIIi. (1888), pp. 147 -172, and, from the aspect of
the theory of the linear differential equations which they satisfy, by Graf, Math. Ann. XLV. (1894),
pp. 235-262; LVI. (1903), pp. 432-444. See also de la Vallee Poussin, Ann. de la Soc. Sci. de
Bruxelles, xxix. (1905), pp. 140-143.