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

19-4]

SCHLOMILCH SERIES

633

Another transformation of the series, also given by Whittaker, is obtained
from the expansion for 1/(1 - e-t) in partial fractions; this expansion is
_1 1 i { I I }
e-1t t 2 + t= t-2m'ti +2m7ri
whence we deduce that
11
(3) 1+ Z e-mJo (tnp) = +
m1 = 1 q
+ E 1
M=J L{(2m7ri + Z)2 + 2 + y2,} V{(2mir - Z)2 + x2 + y2}
It follows that the series represents. the electrostatic potential due to a
set of unit charges (some positive and some negative) at the origin and at a
set of imaginary points.
The reader may find it interesting to discuss the Lipschitz-HankeI integral
of ~ 13'2 as a limiting form of a series of Whittaker's type.
Some other series have been examined by Nagaoka* in connexion with a
problem of Diffraction. One such series is derived from the Fourier series for
the function which is equal to 1/V(1 - x2) in the interval(-1, 1).
The Fourier series in question is
1 co
(4) 1-2x = ~ 2 + Tr Z J0 (rIr)cos mrx,
(l- X2) r=1
and it converges uniformly throughout the interval (- 1 + A, 1 - A), where A
is any positive number.
Multiply by eexi and integrate, and we then obtain the formula (also due
to Nagaoka)
) (1 f ei dx ie 2 1 J0( ) a cos mwrx - mmri sin mvrx
(a) 7( )- 2- a + 2 ', Jo (m7r)
77' \/(I - x 2 a L nz1 a2 - m2W-2
The series on the right in (5) converges uniformly throughout the interval
(-1, 1) and so we may take - 1 and 1 as limits of integration.
Hence, for all values (real and complex) of a,
sin a + 2a ( J (m )
(6) Jo (a)= I + 2a2 - a2
a M=I a;2 -wy r2
A more general result, valid when R (v + 3) > O, is
sinaF (a)" a G (-) J (mr) (
(7) J, (a)= -a [ ( + 2a2^ (J
a r (71 +1) + r M": (a' - 7
~ Journal of the Coll. of Sci., Inmp. Univ. of Japan, iv. (1891), pp. 301-322. Some of Nagaoka's
formulae are quoted by Cinelli, Nuovo Cimento, (4) I. (1895), p. 152.