444 
ME. E. W. BARNES ON INTEGRAL FUNCTIONS. 
sK 1 +• 
+ (- 
n ! 
, — ni - •£ 
i-=0 
a,T (1 + r) 
(- *) r+3 
clz 
if we postulate that we are reversing the process by which we “sum” an asymptotic 
series. 
The integral is equal to the asymptotic expansion 
71 = O' 
S/(l) 
Jl f -na | tT(1 + r) | Sh (1) r | a r V(l + r) 
-VJ C ,.T 0 (_z)r+s-» • n to 7?! L=o F(2 + 
co oo s 7 m r » 
+ 1 S a,w' + 2 -^V J S a. 
■ v __ 
?’=o 1 + r 
r. r — 1 . . . r + 2 — 
n ^ 
J 
f w co 
<f> (on) dm + -k<f> ( m ) + - 
n = 
S'„ +1 (0) d*_ 
dh n + 1! d m 
When n is odd Sb+o (0) = 0 : the integral is therefore equal to the asymptotic 
expansion 
Cm co , o(0) ci? n l l 
j <£ i m ) dm + hf> ( m ) + * o 2) ;Vd / $ M 
= I *< m > dm + W (’») + ,1(2^ 71) • * <”')• 
We 1 lave finally the asymptotic equality^ 
7)1 — l 
V 
71= 1 
</> (n) = Z + j 0 (on) dm - y> (on) + ^w„ + i (m). 
This equality is valid when cf> (z) is a uniform integral function of 2 such that it it 
co 
be expanded in the form ci 0 -f- a x z -j- . . . -fi- o r z r -j- . . . , the series % a,T (I + r ) ~ r 
r=0 
has a finite or infinite radius of convergency. 
_ 2 
We must therefore have X/ a „n ! equal to a finite or zero quantity, so that a ~ 
must increase as fast as or faster than n. The function (f> (z) must therefore be a 
function whose “ order is greater than or equal to unity. 
In the particular case when the series S a r Y(\-\-r)z T represents an integral 
r =0 
function, we may conveniently express Z in terms of the Eiemann £ functions of 
negative integral argument. 
* In a subsequent paper I shall show that it is better to write this formula in the form 
?/? — 1 
2 <f>(a 
71=0 
?u») = 7. + 
v S'„ (a j to) 
i>=o n ! dx n 
nioj 
in order to exhibit its analogy to more general extensions. 
