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

8-32, 8-4]

FUNCTIONS OF LARGE ORDER

241

Again, to prove that dv/dzi does not exceed '/3, we write
_ sin 3 + (v- /3) cos /
sin v
and then it is sufficient to prove that
3+2 ( )-+2 (V)+1 > 0.
Now the expression on the left (which vanishes when v=a3) has the derivate
2+' (v) [3+" (v) - 4, (v)]
= 4' (v [(- 3) {sin2 v + 3 cos2 v} cos 3 + sin2 v sin 3 - 3 cos v sin (v - /)].
sin2 v sin 3 - 3 cos v sin (v - /3)
But (v -i 3) cos 3 +
(v - cs s sin2 v +3 cos2 v
has the posv dri 4 sin4 v cos a s i
has the positive derivate (i2 v+3 2 )2 and so, since it is positive when v=0, it is
(sin2 +4-3 cos2 v)2
positive when 0 < v < 7r. Therefore, since +' (v) has the same sign as v - 3, it follows that
2+' (v) [34~" (v) -+, (v)]
has the same sign as v -, and consequently
3,'2 () - +2 (V) +1
has v=/3 for its only minimum between v=0 and v=rr; and therefore it is not negative.
This proves the result stated.
8*4. The asymptotic expansion* of J, (v sech a).
From the results obtained in ~ 8'31 we shall now obtain the asymptotic
expansion of the function of the first kind in which the argument is less than
the order, both being large and positive.
We retain the notation of ~ 8'31 (I); and it is clear that, corresponding to
any positive value of T, there are two values of w, which will be called Ail and
w2; the values of w, and w2 differ only in the sign of their imaginary part, and
it will be supposed that
I (w) > O, I (w2) < 0.
We then have
ev (eanha-a) X d ' d'
J, (v sech a) = e (a e-a - dr,
2 j7T dT dr
where x = v sech a.
Next we discuss the expansions of w, and w2 in ascending powers of T.
Since T and dT/dw vanish when w = a, it follows that the expansion of T in
powers of w - a begins with a term in (w - a)2; by reverting this expansion,
we obtain expansions of the form
WI- a = " a (m+ (2 'm
wl —a= i+ — 1 1)Y -w-a= E ( f —+-a (W)l a- LT(m+l),
Mn=()4n + n=O mn+1
* The asymptotic expansions contained in this section and in ~~ 8'41, 8'42 were established
by Debye, Math. Ann. LXVII. (1909), pp. 535-558.
w. B. F. 16