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

14-3]

MULTIPLE INTEGRALS

453

integrations permissible. One such set of conditions is that (I should be
bounded as the variables tend to infinity, and that
f (r) = 0 (r-P), ( - 0); f (r) = 0 (r-q), (r ),
where p < 3, q > 1.
A somewhat simpler formula established at about the same time by Weber* is that,
if q (r, 0) is a function of the polar coordinates (r, 0) which has continuous first and second
differential coefficients at all points such that O r < a, whose value at the origin is U0,
and which is a solution of the equation
x- -- + -+ k2=0,
w u
then u/ (r, 0) dO = 2uo Jo (kr),
when 0 S r < a. The proof of this is left to the reader.
14'3. General discussion of Nreumann's integral.
The formula
(1) fuda f" F(R, '). Jo [u N/{R2 + r2-2Rrcos ( )- )}]R (d'IdR)
=2r.F(r, )
was given by Neumann in his treatiset published in 1862. In this formula,
F(R, 1I) is an arbitrary function of the two variables (R, J>), and the integration over the plane of the polar coordinates (R, t) is a double integration.
In the special case in which the arbitrary function is independent of D, we
replace the double integral by a repeated integral, and then perform the integration with respect to P; the formula reduces to
(2) udu F (R) JO (uR) JO (ur) R dR = F(r),
a result which presents a closer resemblance to Fourier's integral}+ than (1).
The extension of (2) to functions of any order, namely
(3) fudu fF(R) J, (uR) J, (ur) RdR =F (r),
0 J0
was effected by Hankel~. In this result it is apparently necessary that v > -2
though a modified form of the theorem (~~ 14'5-14 52) is valid for all real
values of v; when v = + -, (3) is actually a case of Fourier's formula.
The formulae (2) and (3) are, naturally, much more easy to prove than (1);
and the proof of (3) is of precisely the same character as that of (2), the
* Math. Ann. i. (1869), pp. 8-11.
t Allgemeine Lisung des Problemes iiber den stationdren Termperaturzustand eines homogenen
Korpers, welcher von zwei nichtconcentrisclen Kuzgelfiichen begrenzt wird (Halle, 1862), pp. 147 -151. Cf. Gegenbauer, Wiener Sitzungsberichte, xcv. (2), (1887), pp. 409-410.
$ Cf. Modern Analysis, ~ 9-7.
~ Math. Ann. vmII. (1875), pp. 476-483.