INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
269 
The Asymptotic Behaviour of G p (x; 6), when 3ft ( x ) > 0, and /3 tends to Infinity. 
§ 23. The previous asymptotic expansions give us no indication of the behaviour of 
Gp(x ; 6) when /3 grows indefinitely, and 3ft (a;) > 0. For although we have seen that, 
so long as ft is finite, 
where [ J ( x ) | tends to a definite finite limit when | x \ grows indefinitely, yet as /3 
tends to infinity it is possible that | J (x)\ will also grow indefinitely. This possibility 
must now be examined. 
§ 24. We base our investigation on the properties of the function 
S( s )- v _ r ( n ~ s ) _ 
The series is convergent if 3ft(/3 + s)> 0. For, if cr = — s— I, the general term is 
equal to 
, , r(o-+i){ 
r(n + o-+l) _ v J \ 
( 1+ 
b|rH 
1 
b 1 ^ 
M 
L;K f } 
Tln+l)(n+er exp j 
i—* 1 1—* 
+ 
+ ^) 
| (n+oy 
exp 
e -y<7 
n 
r=n+l 
(n+ey 
exp e n 
oo 
n 
(l + -)e " 
'iA) f 
r=n + l 
L\ r) 
nj 
where 9 n tends to zero and n tends to infinity. 
Thus, however large the imaginary part of cr may be (even if it is infinite), the 
series will be absolutely convergent provided 3ft (/3—or— 1) > 0, that is to say, 
provided 3ft (/3 + s) > 0. 
The following argument would have been more simple; it would not however have 
brought out so clearly that the imaginary part of cr may be infinite. 
If u n denote the nth term of the series, 
■ u n+ 1 _ n — g / n+6 V 
u n n+l\n + 9+l/ 
1 — (/3 + 5+ l)/n+(... )/^ 2 + • • • 
Hence, by a known theorem, the series is convergent if 3ft(/3 + s+l) > 1. 
