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

8-51] FUNCTIONS OF LARGE ORDER 255
Again, from (8) we have
e-vF(O, x) r7r
J (vx) - - exp - v0 /(1 - x2)} dO
e-v F(0,x) roo
< f exp {- V62\/(1 - x1)} dO,
so that
e-vF (0, x)
(9) JV (vx) ( _
The last expression is easily reduced to Carlini's approximate expression
(~ 1'4, 8'11) for J,(vx); and so Carlini's expression is always in error by
excess, for all* positive values of v.
The corresponding result for J' (vx) is derived from (7). Write
02 - 2 sin2 0 - G (, x),
and replace G (0, x) by G for brevity.
Then
2xJ' (vx) = 1i e-F(, x) dG (0, x) x
2J x= e- F (o), { x)V dO
Vrdo d1
e-v F (0, x) frX
7r.0
4 exp I- I 7)G/y1(1 + X2)}. GA dG,
and so
(10) xiJ' (rx) < e-YF(, x) (1 + x2){//(2rv).
The absence of the factor /(1- x2) from the denominator is remarkable.
It is possible to prove the formulat
/n x f" ^\x.+1 exp {v /(1 - x2)}
(11) [Jo( dt N
Jv (2t) v3) d(1 _ X2) (1 + (1 X2)}V
in a very similar manner.
This concludes the results which we shall establish concerning a single
Bessel function whose argument is less than its order.
8'51. Lemma concerning F(O, x).
We shall now prove the lemma that, when 0 < x < 1 and 0< 0 < 7r, then
dF (0, x) 8_ X2 sin 0 cos 0
(1) F )- F(, x) - F(O, x) 02_ x in2 ~ 0.
d 0 12s-X sin2.
The lemma will be used immediately to prove an important theorem concerning the rate of increase of J~ (vx).
" It is evident from Debye's expansion that the expression is in error by excess for sufficiently
large values of v.
t Cf. Proc. London Math. Soc. (2) xvI. (1917), p. 157.