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

3-571, 3-572]

BESSEL FUNCTIONS

69

Now, since a is less than 1, 2 (an cos 2n0)/n does converge uniformly throughout the
range of integration (by comparison with 2a'), and so the interchange is permissible; that
is to say
2 oo [ afn cos 2n 2 [ r 0 a, cos 2n0
- /2 cos (z cos 0) da= I cos ( cos 0) 2 d7rn=1J o? 7r o n=l l?
--- - I - cos (z cos 0) log (1 - 2a cos 20 + a2) dO.
7r 0
Hence we have
)J(Z) -= lim - m- cos (z cos 0) log ( - 2a cos 20 + a).
n-=l 7 3 a —l —0 7' J0
We now proceed to shew that*
lim r cos (z cos 0) {log (1 - 2acos 20 +a2)-log (4a sin2 )} dO= 0.
a-l —O 1 o
It is evident that 1- 2a cos 20+ a2 - 4a sin2 0= (1 - a)2 > 0,
and so log (1 - 2a cos 20 + a2) > log (4a sin2 0).
Hence, if A be the upper bound t of cos (z cos 0) I when 0 ~ 0 ~1r, we have
2 cos (z cos 0) {log (1 - 2a cos 20 + a2) - log (4a sin2 0)} dO
~ 0
~ A f {log (1 - 2a cos 20 + a2) - log (4a sin2 0)} dO
Jo
_A f { — 2 +- c S2+log (l/a)-2 log(2 sin0)} d
J0o n = 11 n
=7rA log (l/a),
term-by-term integration being permissible since a< 1. Hence, when a< 1,
cos (z cos 0) {log (1- 2a cos 20 + a2) - log (4a sin2 0)} dO < 7rA log (I/a)-0,
as a — 1-0; and this is the result to be proved.
Consequently
2 (-)' 4(a) = - lin cos (z cos 0).log (4a sin2 0) dO
n —=? a-l-.0 7 J O
=- 1 jf cos (z cos 0). log (4 sin2 0) dO,
7r 0
and the interchange is finally justified.
The reader will find it interesting to deduce this result from Poisson's integral for J, (z)
combined with ~ 3'5 (5).
3'572. Stokes' series for the Poisson-Nreutmann integral.
The differential equation considered by Stokes ++ in 1850 was -d + dY - m2y, where
m is a constant. This is Bessel's equation for functions of order zero and argument imz.
Stokes stated (presumably with reference to Poisson) that it was known that the general
solution was
y/ =1 {C+ D log (z sin2 0)} cosh (mz cos 0) dO.
* The value of this limit was assumed by Neumann.
If z is real, A=1; if not, A <exp {| I(z) z}.
+ Trans. Camb. Phil. Soc. IX. (1856), p [38]. [M38] athematical and Physical Papers, Im. (1901),
p. 42.]