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

1*6] BESSEL FUNCTIONS BEFORE 1826 11
When k is large, 1/(4k2) may be neglected in comparison with unity and so we may expect
that J0 (k) J/k is approximately of the form A cos k + B sin k where A and B are constants.
To determine A and B observe that
cos k. Jo (k) - sin k. Jo' (k) =1 f {cos2 Ico cos (2k sin2 ICo) + sin2 Io cos (2k cos2 ~I)} i)co.
r 02 2
Write r - co for o in the latter half of the integral and then
cos k. Jo (k) - sin k. Jo' (k) = 2 cos2 co cs (2k sin2 0) d
2/2 / (k)/ X2\2
/ f\ ) ( - x2) cos x2dx,
2rr^,1 - COSi/ X
and similarly sin k. J0 (k) +cosk. Jo' (k)=- ( - sin x2dx.
7r /ky o \ 2k/
But lim f (2) (12k csx.d c.d^ ),
k- jo 0 2kJ s in o sin /
by a well known formula*.
[NOTE. It is not easy to prove rigorously that the passage to the limit is permissible;
the simplest procedure is to appeal to Bromwich's integral form of Tannery's theorem,
Bromwich, Theory of Infinite Series, ~ 174.]
It follows that
(cosk. J0 (k)- sin k. J0' (k)= 1k (1 + S),
(1 +rk),
Isin k. J (k) + cos k Jo' (k) = -(,k (1
where Ek. —O and k-0 as k —no; and therefore
J (k) =, j) [(1 + sk) cos k + (1 + k) sin k],
J0 (k) /(k) [-(1 +Ek) sin k+(l +~k) cos k].
It was then assumed by Poisson that Jo (k) is expressible in the form
/ [(A+ -+ + -...) cos k + B++... ) sink,
V(rk) k2 1 J
where A = B= 1. The series are, however, not convergent but asymptotic, and the validity
of this expansion was not established, until nearly forty years later, when it was investigated by Lipschitz, Journal fiir Math. LVI. (1859), pp. 189-196.
The result of formally operating on the expansion assumed by Poisson for the function
d2 1
J0 (k),/(7rk) with the operator dk+ 1 +4~ is
-COSk[2. 1.l B - 4A 2.2B"-(l.2+)A'+ 2.3B -(2. ) A.+sin[2.. A' +;B 2.2A"+(1.2 +)B' 2.3A'1 +(2.3+) 1B"
+siln k+ -- -+ 4+ J>
k2 V
* Cf. Watson, Complex Integration and Cauchy's Theorem (Camb. Math. Tracts, no. 15, 1914),
p. 71, for a proof of these results by using contour integrals.