442 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
With our previous notation we take b m = and 
1 ml 
m= 1 
7T 2m F 2 m ( ~ %) 
ml (sin 7 rfl) 2m 
2 (~xf 2 — 
r=0 m=l m- ! 
7T 
1 ! sin 7 t6 
2 m 
I- CO 
= s (-xy 
r =0 
( -7 -Y~ 
/j \r ! sin 7 -0/ _ 
so that f {%) is a transcendental integral function. 
Then we have f fix) (- x) n+e ~ 1 dx = 2 ^ r(n + g) ^'" = e [m + o? _ L 
2ir J ^ v v 7 m= i m ! 
We take the associated function 
X (z) = c 0 + CjZ + . . . + + • • • where c~ l = e [r(ft+0)]J — I ; 
and the integral function 
c 4r('/i)P 
G ( 2 ) = a 0 c 0 + . . . + a n c n z" + . . . 111 which L t a n c n = L£ ■ [r(?t+a)p = 0 . 
Then the sum of the series will be represented by # 
| G ( — xz) f(x) (— x) e ~ l dx. 
§ 40. We have now, by means of the generalised exponential functions, given the 
machinery by which we may expect to be able to “ sum ” a natural asymptotic series 
of any order. 
It may be proved just as for the fundamental exponential process that the series 
and the function derived from it have asymptotic equality of the arithmetic type. 
Moreover, if we regard the series as having a finite radius of convergency, on 
which one or more singularities lie, which has shrunk indefinitely, we, as it were, 
magnify it again by means of the function F„ ; (x) so as to obtain the associated 
function 
00 
G (u) = 2 (— ) nm a n c n u n 
71=0 
whose radius of convergency is infinite. 
The alternative process consists in successive magnifications by means of the 
function e v . 
These two jirocesses will in general lead to different results : in each case we shall 
obtain functions with 2 = 0 as an essential singularity; both functions will have the 
, -1 
* When we take b m = (rn !)», we have 
f(x) (-x) 
i 
2 k 
[r (n + fl)] 2TO . 
n+e - 1 dx= 2 7 x 
m =1 (m!) ,/s 
and, when n is large, the series is, by a theorem due to Stokes, infinite like exp. <j^ J- [T (?j + tf)]*} 
to the first approximation, We thus sum any series for which a n is of order exp. {[T (»)]''}> by taking 
s greater than r. 
