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

18-22]

FOURIER-BESSEL SERIES

587

Now take 0 < x <1. in the last integral and substitute for the Bessel functions
the dominant terms of their respective asymptotic expansions, valid when w I\is large (~ 721). The error produced thereby in the integrand is, at most,
O (1/w2) when 0 < x < 1; and, as n"- oo, we have
rAn+ ci dw. 0 \
Jnn-oi W2 An~, \).
Now the result of substituting these dominant terms is
QXv /An+i sin w (1 - x) sin (xw - - ~- ir)
- lim - - _-2. -dw
B B- ao JrB w c Ano-Bi w COS - - 1 -4)
- v A * +Bi_ [I cos (2w - - PV - C 2)
l m- di v. i -- CvW
_ v_ lim |An+Bi cos (2xw - w - - 7r):
BG- J A,-Bi W COS (W- 2 vT-4 d)
We shall have to discuss, almost immediately, several integrals of this general
type; so it is convenient at this stage to prove a lemma. concerning their
boundedness as n - co.
LEMMA. The integral
lr An+ Bi cos (Xw -vr- 4r) d
lim — dw —2. —/r
B-co JAn-Biw COS (W- -41r)
isO.(1/n), as n-co, if - 1 < X < 1; and the integral is bounded if 0 < X 1.
If we put w-An ~ iv, where An, as usual, stands for (n + iv + )) 7r, the expression under
consideration may be written in the form
co (X - A. -sinf [* (cosh XXv. dv Sih X..dv
2i An cos (\-1) An. 2 — A) d- -sin (X-l). -( 1 2Alt..LA 1) J coshc( - 1hv (. o A-+-i ohvv
When - 1 < X < 1, the modulus of this does not exceed
2 fo cosh Xv.dv _2 [ v sinhX vl dv." cosh,v + '
An Jo osh v o cosh v cosh
and the first part of the Lemma is obvious.
Again, if 0 6 X 1 and v ( - X) =, we have '
0 v (1 - X) sinh Xv = ~ sinh (v - $) 6 cosh v,
so the integral to be considered does not exceed (in absolute value)
Ak, lo 1 +2A o An + v2-r,
and the second part of the Lemma is proved.
It follows immediately from the Lemma that
fty+lTn (t, x) dt = xv + 0 (1/n),
Jo
when 0 < x < 1; and this is equivalent, to (2).
* The function t sinh (v - t) has one maximum, at 50 say, and its value there is equal to
sinh2 (v - to) cosh (v - W0) which is less than sinh(v - -o).