MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
443 
same arithmetically asymptotic exj)ansion. But the expansion in all probability will 
not be valid in the two cases along the same lines or within the same areas tending 
to 2 = 0. Moreover, such a result is not surprising. The original series, except 
from the point of view of the computer, had no meaning ; it did not define an 
analytic function over any area of the plane of the complex variable and therefore 
could not uniquely represent such a function. We have, however, now given two 
processes (out of an infinite number) by which we may conceive the series to define 
an analytic function, and the functions thus defined each satisfy all that the 
computer can demand. 
§ 41. It will, perhaps, elucidate the theory which has been developed if we give two 
actual examples of its application. 
We will first investigate the Maclaurin sum formula, which gives an asymptotic 
m— 1 
value for X <f> (n) when m is large, under certain restrictions as to the nature of the 
n=l 
function (f> ( n ). 
In the first place it is evident that such restrictions must exist : the function must 
either be uniform or be limited to a definite branch of a multiform function; and, as 
2 takes increasing integral values, (f) ( 2 ) must increase uniformly. 
We will assume, therefore, that <f> ( 2 ) is an integral function, which may be 
represented by a Taylor’s series, a 0 + apz + . . . + a r z r -j- . . . 
Then, if the integral be taken along a contour embracing an axis in the positive 
half of the 2 plane, we shall have, by the usual expression for the gamma function, 
m m . C oo 
t <f> (n) = % — Hr(l+r) e~ nz (- z)~ r ~ x dz = 
71=1 71=1 — ^ J r= 0 
i 
e 2 
1 — e mz - a r T (fo r) 
i-«-*r=o (-yr +1 c: 
Suppose now that the series X a r T (1 + r) z r has a finite radius of convergence p. 
r=0 
CO 
Then 2 a r Y (1 + r) (— 2) _r_1 will be the expansion of a function convergent 
r=Q 
outside a circle of radius p~ l . 
We can always make the bulb of the contour along which the fundamental 
integral is taken expand so as to entirely include this circle of radius p~ l . and the 
subsequent integral will then be finite. 
i f €~ z co CL t\ 
Let now Z = 0 ——— X - —Wqd 2 , so that Z is a definite finite quantity 
"7T J 1_ 0 t=0 \ 
depending on the coefficients in the expansion of 6 ( 2 ). 
„-(m+i)z x a r r (1 + r) ^ 
Then X (f) ( n) = Z — ~ [ 
n= 1 47T J 
1 - <T 
r—0 
(-2W 1 
The second integral may be written in the form 
3 l 2 
