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

634

THEORY OF BESSEL FUNCTIONS

[CIAP. XIX

JY (a)
This expansion is also obtainable by expressing a as a sum of partial
a" sin a
fractions.
Various representations of the integral on the left of (5) were obtained by Nagaoka;
the formula quoted seems to be the most interesting of them.
Finally we shall give the formula t
(8) J Jo{(2m- 1) x}j 7r2 _ <
=(8 (2m - 1)2 8 2(- < )
This is deducible from the Fourier series
c cos(2m-l)- 77- _ s i) ( < )
adz= (2m-^ = r - 2 x ) (-r < 7r)
m,=t (2m-_)2 8
by replacing x by x sin 0 and integrating with respect to 0 from 0 to -Tr.
As an example of the calculation of the sum of a Schlomilch series when
the variable lies outside the interval (- 7r, r-), we shall take 7r < x < 2vr, and
then, iff(x) denotes the sum of the Fourier series, we see that
J Jo{(2m- l)xl 2 = ^
m=1 (2 1)2 r /(xsin d ) dO
M=1 (2m -l) - rJo
2 f are sin (7rf/) f ir r
=2- jE+ f (x sin 0) dO
7r ( J are sin (7r/x)
9 rarcsin (ir/X) rr
I= (7r- 2x sin 0) dO
7/' O 8
+ 2,r/ ) (2x sin 9 - 37r) dO,
7r arcsin (7r/x)
so that, when wr < x < 27r, we have
(9) _ Jo(2m-1)= ( _r2)-x -7rarc cos +?(9) m=1 (2m - 1)2 V2
19'41. Null-functions expressed as Schlomilch series.
We shall now prove the remarkable theorem that
(1) 1+2(-2JoM ^O,~Oo
2 m=1
provided that O < x:< 7r; the series oscillates when x= 0 and diverges to
+ oo when x= rr.
This theorem has no analogue in the theory of Fourier series, and, in fact,
it is definitely known+ that a Fourier cosine-series cannot represent a nullfunction throughout the interval (0, 7r).
* Cf. Modern Analysis, ~ 7'4.
t This was set as a problem in the Mathematical Tripos, 1895.
+ Cf. Modern Analysis, ~~ 9'6 —9632.