448 
MR, E. W. BARNES ON INTEGRAL FUNCTIONS. 
integral functions, we now proceed to sketch the process which will be adopted, and, 
in the course of our outline, to prove at once the validity of that process and the 
laws which govern its results. 
Suppose, in the first place, that we have the absolutely convergent expansion 
F ( 2 ) = a 0 + cqz + . . . + o s z s + . . . in which the coefficients are functions of a 
variable t, asymptotically given for large values of j 1 1 by expansions of the type 
a n — w; + pg + ^7 H - • • • where the quantities b n _, b ni , . . . are constants and n 0 , n 1} . . . 
are numbers arranged in ascending orders of magnitude and tending to + co as a 
limit, the first numbers of the series being possibly negative. 
Suppose that we substitute these asymptotic values of the coefficients and 
rearrange the expression for F ( z ) in powers of 
t 
We shall obtain, when U | is large, an asymptotic equality 
F («) = ¥ , 
K + ^ 10 2 + • ■ • + + • 
+ t -, h 
K + b n Z + . • • fi- + • • • 
+ • • • + NT 
hs + b ls z + . . . + b ms z m + . . . 
4- • • • 
This expansion will be arithmetically asymptotic : the computer would use it to 
calculate F (z) for given values of 2 and t when \t\ is large. 
The series which enter as coefficients will be, in all probability, divergent; but, as 
we are looking at the whole matter from the point of view of the computer, we are 
at liberty to “sum” them by the methods which have been developed in the present 
part of this memoir. 
If, as will be the case in the applications which we subsequently make of this 
theory, these series have a finite radius of convergence, we can “sum ” them each to 
a definite, possibly non-uniform, analytic function ; and we shall have an expansion 
co 
F ( 2 ) = S 
s =0 
dependence. We shall thus have obtained a unique asymptotic expansion for the 
function F ( 2 ). The case in which the series of the type 
/,(*) 
w 
liich will satisfy Poincare’s definition of arithmetically asynqitotic 
b QS + b ls z + . . . + b MS z"' + . . . 
have zero radius of convergence does not arise. I 11 such a case we should be able to 
obtain an infinite number of functions, of which these series are the asymptotic 
expansions, and we should have the absurdity that the asymptotic expansion of F ( 2 ) 
in ascending powers of — is not unique. 
i 
§ 44. A function cannot, as has already been stated, be represented by the same 
asymptotic expansion for all values of 2 in the neighbourhood of 2=00 , unless the 
function is an integral function of z~ 1 , and the series absolutely convergent. 
