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

7.32]

ASYMPTOTIC EXPANSIONS

209

7'32. The signs of the remainders in the asymptotic expansions associated
with J, (x) and Y, (x).
It has already been seen that J, (x) and YI (x) are expressible in terms of
two functions P (x,v) and Q (x, v) which have asymptotic expansions of a
simpler type. We shall now extend the result of Stieltjes (~ 7'31) so as to
shew that for any real value* of the order v, the remainder after p terms of
the expansion of P (x, a) is of the same sign as (in addition to being numerically
less thant) the (p + i)th term provided that 2p > v- a corresponding result
holds for Q (x, v) when 2p > v - 3. The restrictions which these conditions lay
on p enable the theorem to be stated in the following manner:
In the oscillatory parts of the series for P (x, v) and Q (x, v), the remainders
are of the same sign as, and nzmerically less than, the first terms neglected.
By a slight modification of the formulae of ~ 7'3, we have
P vx-p) >2 jj+1) fe-2t uv-t {(1 + ~iu)v1 + (1- i)i} d',
'(v+ )Jo
Q V+, v12 1 ~.
Q(,) =2ir(v+1)0 + I 2{( t+i r- - (I -1 t ~)V du,
and, exactly as in ~ 7'3, we may shew that
p-i ( )r /1 -,1\1)2m
2{(l + i )- + =(1- ) } = ( - ) (- (2)
m=o (2m)!
+ (-)C P (~ _ ( ))P,1f () - t)1 P- -i (1 + ilt)v-2P+ - (1 - iUt)v-2P+} dt.
(2p - 2)! 2o
The reader will see that we can establish the theorem if we can prove that,
when 2p > v - ~, the last term on the right is of fixed sign and its sign is that of
(-)P. ( -^ )2p (I)/(2p).
It is clearly sufficient to shew that
1 1
2 1 [ - (1 - t)2p-2 i {(1 + iut)v-2+ - (1 - ~iut)V-2+} dt
2p - v- o
is positive. Now this expression is equal to+
(2p - 1) F (2p- P- 1) (1 - t)-2.i - te-Ax (+itt)_ -e-x (1- iu}t) d dt
= (2p 1- + )jj ( - t)2P-2 X2P-V- sin (~X t). e- dxAdt
F(2p_ + +1). 2
1 (2p-v + ) +-e mo t- e-j ( - t)2P-2 sin (1 Xut) dtdX.
* As in ~ 7'3 we may take v O without loss of generality.
+ This has already been proved in ~ 7'3.
+ Since I sin ( Xut) ] <ut, the condition 2p>v- secures the absolute convergence of the
infinite integral.

W. B. F.

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