466 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
But this integral is equal to 
(—)*<£* O) 
log <j> (m) + 
* 
N' 
s=-l- 
= log M - log 
1 + 
0 ('»<)" 
+ log 
1 + 
<M W )_ 
= log 
Thus, where e 2 = 1, the difference of the two asymptotic approximations commences 
with the term — ?qp _1 log 2 , a result which may be obtained without much difficulty 
by elementary algebra. 
Note that — 6 } p _1 log s is the expression obtained when we reject the infinite part 
— bir z e 
of the function —-—.— fl for 0 =0, when expanded in powers of 0 . 
p sin 7r 1 1 
■ Note also that the constant Z ( 0 ; f v ‘ ’ ’ which, when e, > 1 is the first term of 
V K »■>•••/ 
the asymptotic expansion of the logarithm of the ratio of our two products is 
cc q r. 
equal to % log - p . 
71= 1 
By means of the formula x > log (1 + x) > ^ ^ , where x is a real quantity lying 
between ffi 1, we may prove that this series is absolutely convergent when p > 1. 
Application to Functions of Zero Order. 
§ 60. Hitherto no example has been given of a function of zero order, although the 
general investigation of § 36 applies equally to functions of this nature. In such 
cases it becomes necessary to introduce Maclaurin constants of a complexity which 
seems, except in special cases, beyond the reach of present analytical processes. They 
can no longer, as for functions of finite order, be expressed in terms of IIiemAnn’s 
£ function nor, I believe, in terms of any functions which have so far been introduced 
into analysis. An example will now be given of a very rapidly converging integral 
function. It obviously would serve as the starting point of a series of interesting 
researches dealing with the classification of simple integral functions of zero order. 
61. We propose to obtain the asymptotic expansion of the function IT 
)i=i 
In the notation of the general theory we have now 6 (u) = e‘\ 
Therefore log (n) = n ; % log (fi (n) = — — — . 
By the Maclaurin sum formula, if 5 be positive, 
m— 1 rm pus p> 
N. e' ls = j e ns cln + Q, — — + ^ sc ms — . . . 
where C s is the Maclaurin constant corresponding to e“ s , which may be determined as 
cm 
in § 84. If we put e"\ln = e m )s, we have C, = (1 — c j _1 . 
