INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
253 
In conclusion, I must mention a paper of Hardy 1 in which he obtains some of the present results. 
He was led to the question of asymptotic behaviour by a desire to obtain approximations for the large 
zeros of integral functions, one of the subsidiary problems which a general knowledge of integral functions 
will solve. His paper was sent to me in August, 1904, in the capacity of referee to the London 
Mathematical Society. He had obtained the first terms of the asymptotic forms of the function which 1 
call Gp (x; 6) in the case where and 0 are real. In my reply I said that I had already obtained complete 
expansions for general complex values of (3 and 6. Such results Mr. Hardy has since published in the 
revised form of his paper. The reader will find it instructive to compare our respective discussions of 
the question. 
[Note added March 21, 1906.—The Council of the Royal Society suggested that the paper in its original 
form contained so many developments that it was more of the nature of a treatise than of a paper to be 
published in their Transactions. In consequence it has been considerably compressed, and statements of 
results have been in many cases given in lieu of detailed investigations. Developments of such a nature 
will, I hope, with my subsidiary investigations, be suitable for publication elsewhere. In compressing the 
paper, ceitain changes have at times been made in the mode of presentation. Whenever such a change 
has been made, or whenever a result has been stated which had not been originally obtained, the number 
of the corresponding paragraph is placed in square brackets [ ].] 
Preliminary Definitions and Theorems. 
3. The function f(x) is said to admit the asymptotic expansion 
Ci Co 
G>+ - + - 2 + ... 
X X w 
for a given range of values of arg x, when \x\ is large, if the following condition is 
satisfied. We put 
f{x)—c 0 — — — 
x x 
= R, 
then it must be possible, for any assigned value of n, to find a value X for |x| such 
that, whenever 
M>X, \x n R n \<e, 
where e is any arbitrarily assigned small positive quantity. 
The solution of linear differential equations often gives rise to series of which the 
simplest type is 
<*„+- 
X X 
the series within the brackets beino- divero-ent. 
O © 
We say that f («c) is asymptotically represented by such a series for any value or 
range of values of arg x under the following conditions. 
Put 
f(x)—e x 
c 0 +- + ...+ f 
X 
X 
1 Hardy, “ On the Zeros of Certain Classes of Taylor’s Series,” Part II., ‘ Proc. Lond Math. Soe.,’ 
ser. 2, vol 2, pp. 401-431, 
