MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
453 
.supposed to apply. We shall call y s the Maclaurin integral-limit for f (n). We 
shall also put F* = — ( <f>* (n) dn, and call F, s the 5 th Maclaurin constant for <f> (n). 
When s = 0, we have the formula 
to — 1 Fm 
t log 4 > (n) =| log 4 > (n) dn — \ log <f> (m )... + ( — Y 
n= 1 - 1 y 0 
B, +1 d )** 
(21 + 2)! clm 2t+1 
log ... , 
where y 0 is the Maclaurin integral-limit for log c/j (n). 
^ f y ° 
We put log F 0 = ) log <f> (n) dn, and call F 0 the absolute Maclaurin constant for 
<t> ( n )- 
When 5 is a positive integer and L t \_ 4 >~ s (n)j = 0 , if is evident that y_ s = — oo 
n= go 
and F_. ? = 0. 
§ 49 . In the particular case when <f> ( n ) = n p , p being real or complex, the 
Maclaurin constants are particular cases of Riemann’s £ function. 
For, for all values of s. 
JL 5 / — s\ 
TO-1 1 1 
n s ^ ^ (1 — s)m s ~ 1 2 m s Fi V 2t ) (s + 2 t — l)m s+3<_1 
When s = 1 , we have L t £ (s) . 
8=1 L 
m=co 
We have also the special values 
C(o)= -b 
m 
1 s 
log 
m 
r 
s! 
£ (.§) = /' B^ +1 , when s is an even positive integer, 
£ (s) = 0, when s is an even negative integer, 
£ (.s) = — —--- 1 , when s is a negative odd integer equal to — ( 2 t + 1). 
— t “f” 
We write, when s is any quantity real or complex, £( — s) = F (s), unless s = 1, 
in which case we put y = F (s). 
Simple Integral Functions of Finite Order Less than Un 
§ 50 . Before we proceed to consider the general theory of the asymptotic 
expansion of functions typified by P p (z) = II 1 + ~ p , where p > 1 , we will 
n=l L 70 _ 
