INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
275 
We have R „ 
(;x ; 0) = 2 b r G p+r (x; 0) + 2 b r G p+r (x ; 0). 
r=0 J-.-R+1 
Consider the second series. However large r may be, we may put 
\ e ~’ b ’ G f*’( x > e )\ = J^W-y 
where, by § 28, e, can be made as small as we please by taking | a: sufficiently large. 
Hence 
2 b r G p+r (x ; 6) e V +R | < 2 e R+r <^, 
r=R + l 
V 
r=l 
where y can be made as small as we please by taking | x \ sufficiently large. For we 
have only to take | x | so large that the quantities e r form an absolutely convergent 
series. 
Again, by the asymptotic expansion for G p (x;0), when | arg x | < 7 t/ 2 , we have 
r,X R 
t b r G p+r (x; 0) = - 2 % 
r=0 
| r c„r(l-/3-r) I N (x) 
** -r/1 n \ * at 
x p r= o x r l_)i=o r (1 — fi—r—n) x 11 
x 
where the coefficients r c n are defined as in the enunciation of the present proposition, 
and where |I N (»)I can be made as small as we please by taking \x\ sufficiently large. 
We therefore have 
R 
2 
r =0 
2 b r G p+r (x ; 6) = — 
v L x b mm Cs-m~r (1—fi—m ) J„{x) ' 
xP\j,=oafm =0 r(l-(3-s) x* 
• (A), 
where cr < N and | (a;) | can be made as small as we please by taking | x \ sufficiently 
large. 
If, now, we take R><x and combine the two results just obtained, we have the 
proposition stated. 
§ 32. The reader will notice the far-reaching generality of the function whose 
asymptotic expansion has been obtained. He will also notice that we have shown 
that from the asymptotic expansion of each of a convergent series of functions we 
have derived an asymptotic expansion for the function represented by the series of 
functions. 
§ 33. Let us noiv consider the asymptotic expansion of F„ (x ; 0) when E (x) < 0. 
We have seen that 
oo 
Fe (x ; 0) = 2 b r G p+r (x;6) .(l). 
r =0 
Also, when |arg( — x) \ <7t/ 2, we have the asymptotic equality 
Gp +r (x; 6) = (—x) 0 {log(-x-)}^ +r41 { 2 
()»r (n) (<9) 
n =oT(l3+r—n) r(n +1) [log(—a?)]” + [log (—a:)] 
An (a) 
IN f ’ 
where |An(^)| tends to zero as |cc| tends to infinity. 
2 n 2 
