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

6-23]

INTEGRAL REPRESENTATIONS

183

From these results we see that
(12) 2 cos 1 v7. K ) - e-ixsinht cosh vt. dt,
~ -00
so that
(13) K, () cos (x sinh t) cosh vt. dt,
and these formulae are all valid when x > 0 and -1 < R (v) < 1.
In particular
(14) K (x))= cos (x sinh t) dt = c (t)t.0o~(t2+ 1)
a result obtained by Mehler* in 1870.
It may be observed that if, in (7), we make the substitution ~zet=r, we find that
(15) IK, (z)= - (z)v l exp { —4 r- +,
provided that R (z2) > 0. The integral on the right has been studied by numerous mathematicians, among whom may be mentioned Poisson, Journal de l'Ecole Polytechnique, Ix.
(cahier 16), 1813, p. 237; Glaisher, British Association Report, 1872, pp. 15-17; Proc.
Camb. Phil. Soc. II. (1880), pp. 5-12; and Kapteyn, Bull. des Sci. Math. (2) xvi. (1892),
pp. 41-44. The integrals in which v has the special values I and - were discussed by
Euler, Inst. Cale. Int. iv. (Petersburg, 1794), p. 415; and, when v is half of an odd
integer, the integral has been evaluated by Legendre, Exercices de Calcul Integral, I. (Paris,
1811), p. 366; Cauchy, Exercices des Math. (Paris, 1826), pp. 54-56; and Schl6milch,
Journal fiir HMath. xxxII. (1846), pp. 268-280. The integral in which the limits of integration are arbitrary has been examined by Binet, Comptes Rendus, xII. (1841), pp. 958 -962.
6-23. Hardy's formulae for integrals of Du Bois Reyrond's type.
The integrals
sin t. sin - dt, cos t. cos -. tv- dt,
Joslntsio X;2t o t
in which x > 0, - 1 < R (v) < 1, have been examined by Hardyt as examples
of Du Bois Reymond's integrals
f /(t) s t. tv-l dt,
cos
in which f(t) oscillates rapidly as t -- 0. By constructing a differential equation
of the fourth order, Hardy succeeded in expressing them in terms of Bessel
functions; but a simpler way of evaluating them is to make use of the results
of ~ 621, 6-22.
* Math. Ann. xvIIT. (1881), p. 182.
t Messenger, XL. (1911), pp. 44-51.