INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
281 
We shall also show that however large E (p) may be, if m be any finite quantity 
: p, we may put 
0 F 1 {p; y } = e 2 
^ AM 
.(m— l)/2 
y 
-where, by taking \y\ sufficiently large, we may make | I p (y) | as small as we please; 
and where \ I p (y)\ tends to a finite limit as p tends to infinity, \y\ remaining constant 
We now put 
(— ) m r (m) (0) 
L , \ + 
m ! (m + 1) ! l ’ 
<Hy) = + s 
_m=0 m=R+l 
The first series in which the summation is taken from 0 to R is equal to 
o2Vy 
R 
V 
(-)T<->(0 r(2m+3 
_m=o m ! ( m + 1 )! F (m + 3/2) 
— to— 3/2 R-m+1 
Woj w + 3/2, -l/2-m;—\= 1+AM 
1 ' Wy\ y ni2+z,i \ 
where Wo {«/} denotes that the sum of the first h terms of the series in ascending- 
powers of y is to be taken. The modulus of J x (y) can be made as small as we please 
by taking | y | sufficiently large. 
The second series 
i (yOUU) oFi{M+2;2/} 
m=E+i m! (m+ 1)! 1 J J 
is equal to 
e w» l (-)T'-»>(0 I„, (y) 
!»=r+i m \ (w +1)! y< R+2 )/ 2 ’ 
wheie | (y) | for all values of m can be made as small as we please by sufficiently 
increasing | y |. 
_ 00 /_ \m j-i(to) /m 
® mce ^ ~ * s absolutely convergent, we see that the second series may 
'i=o m ! (m+ 1)! 
be written 
o 2 *jy JgQy) 
y 
,(R+2)/2 ’ 
when | J 2 (y)| tends to zero as \y\ tends to infinity. 
Finally, therefore, if E (x) < 0 and | arg (—as) | < tt/2, f(x) admits the asymptotic 
equality 
(-x) 9 exp{2y'Tog(-*)}{4v/log {-x)}~ 3/2 
Wo fm+ 3/2 , -\-m ; 
(2m+ 2 )! 
Li. 
R-m+q 
+ 
4 v / log(—x)i {log (—a?)} E/2 _ 
where | J (x) | can be made as small as we please by taking | x | sufficiently large. 
§ 40. W e have therefore obtained the nature of the asymptotic expansion of f(x) 
for the two cases when E (x) > 0 and when E (x) < 0. 
Ihe integral which was employed in § 35 showed us that in the latter case \f(x)\ 
when \ x \ is very large is of order less than the order of \(-x)-°\ |af| when e > 0. 
We readily see that this is in agreement with the preceding result. 
VOL. CCVI.-A. 2 O 
