MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
489 
And, since A is given by the identity, 
in —1 
2 n log n = log A + 
n = 1 
m- 
ITt \ r ¥T$ > 
— ~ -f- jA j log m — — + terms which vanish when m is 
infinite, we have log A — — £' ( — I) + yy- 
Thus the asymptotic expansion of the present paragraph may he written 
T 2 - log A + * log 2 tt + -~h) log 2 ~ y 
+ 
(-) S B, +1 
s=i 2s . 2s + 2 2 3s ’ 
We thus obtain a valuable verification of our results. 
Repeated Simple Integral Functions of Transcendental Index. 
83. It is obvious that such a function as 
n 
k=i 
is of infinite order when a n is algebraic. 
If, however, a n is of the same order as e n , tlie order of the function is finite, and can 
therefore be expanded in the neighbourhood of infinity by our methods. 
We shall take, as an example of repeated functions of transcendental index, the 
product ' 
GO 
n 
71 =1 
& n 
n n £ 
£ /IV' 
,=A «v 
This function is of order p, greater than or equal to unity. 
Suppose first that p is not an integer, so that p is the integer next greater than p. 
Then without former notation 
VI — 1 
^ e Mn —• 
ii=i 
ra— I 
2 e mn qn = 
v. = 1 
fm e n n 
| e mn dn + M — — + ... 
f m 
e vqn qn dn — log F 0 — \ e mm qm + . . . 
And 
m— 1 rr, 
2 q'M' 1 + s 2' ! — 
rt = 1 J 
l pp+s qm 
e (p + s)an dn _Y s — ~ - - + . . . 
r , . log z m ,1 , I. 1 f( — )- s_1 e? + 
f (z) = —— e Pf?W! — — eJ'i m A - e pim + 2 
' m P FI .=-t2 L 
5 qm 
p + s 
where we take the limit when k — co of the summable divergent series. 
VOL. CXCIX,-—A 3 R 
