ME. E. W. BARNES ON INTEGRAL FUNCTIONS. 
431 
just considered. Instead of the process of summation leading to the same result, 
whatever the nature of the integral process chosen, we can obtain an infinite number 
of results, each associated function leading to a different function, of which the given 
series may be regarded as the asymptotic expansion. For when the divergent series 
is convergent for sufficiently small values of | z |, it defines a function over that area 
of convergence, and any summation process can only lead to the analytic continuation 
of a definite branch of that function. But a true asymptotic series has no area of 
convergence, and any meaning which we care to attach to it will harmonise with 
Weierstrass’ theory of functions. 
The essential nature of the difference between the two kinds of series may be 
brought out in another way. A series convergent for sufficiently small values 
of 1 2 : | represents a function regular in the neighbourhood of the origin. But any 
function which a true asymptotic series can represent will have the origin as an 
essential singularity. And, therefore, not only can many functions with an essential 
singularity at the origin have the same asymptotic expansion, but also the same 
function may have different asymptotic expansions in different areas having the 
origin as apex. It is almost impossible to imagine a vagary which an essential 
singularity will not possess, and this fact we cannot, throughout the whole of the 
investigation, too carefully bear in mind. 
Inasmuch as any means of regarding an asymptotic series leads to a result peculiar 
to that means, we must choose our process with care so as to obtain the most simple 
result, and, if possible, so as to ensure that our conception of such series agrees with 
the arithmetic point of view by which historically they were generated. 
§ 29. Suppose, in the first place, that we have given the asymptotic series 
CIq -j- CLyZ -f- cgz 2 -}-... -f- ct n z n 
_ n/ 
in which, by Cauchy’s rule, L t = 00 • And suppose further that Lf ALA 1 — 0 . 
71 = 00 71 — co 
Then the associated function 
G (z) — « 0 c 0 4~ ct\C x z + . . . -\-a n c n z“ + . . . 
in which c n = 
, will be an integral function. 
r> + 6 ) 
It is a natural extension, then, of our previous ideas to regard the asymptotic 
series as the expansion of the integral 
1 
2 sin 7 t6 
| G (xz) e~ x (— x) e 1 dx. 
and, conversely, to regard the integral as the “ sum” of the asymptotic series. 
