ME. E. AY. BARNES ON INTEGRAL FUNCTIONS. 
481 
Simple Repeated Functions of Finite Order less than Unity. 
§ 76. The most general function of this type may be written 
F (z) = n 
?i=i 
1 + a. 
where the principal value of each term is taken when y„ is any function of n ; and 
where a n is a function of n which increases without limit as n increases, and which is 
oo 
such that 2 yfa,, is absolutely convergent. The function is of finite (or zero) order 
n= 1 
less than unity ; and, when y n is an integer, its n th zero is repeated y„ times. 
We take 
a n = (f) ( n ). 
When r = <£ ( n ) we suppose that inversely n = */;(/'). Suppose that z = Re' 9 , then 
if we take m to be a large integer such that m — 1 < \ft (R) < m, we have 
m— 1 
m— 1 
iii — 1 
n= 1 
n= 1 
n = 1 
log F (z) = 2 y„ log z — 2 y„ log <£ (n) + 2 y„ log ( 1 -f - ) + 2 y„ log ( 1 + 
a. 
We carry out our analysis in a manner which depends exactly upon the argument 
previously employed in the corresponding case for non-repeated functions. 
We have at once, in the limit when h — «, 
rn— 1 
m— 1 
n tt 1 / \ i v ~\ j/\, ’V i (—y i y„F( n ) , o k (—y 
log F (z) = log z 2 y H - 2 y„ log (n) + 2_ 2 -—- + 2 2 s .^ (/ , ) • 
11 = ] s=l 
n=m s=l 
Now, if .s be positive, 
in— 1 pth 
2^ y„(f> 3 (n) = I y„f(n) dn - byj/ (m) ... + (-) ^ + 2 , 
B m d 2<+1 . . 
- y m <f> s (m) + 
where y s is a constant depending on s and on the forms of y n and <f> ( n ). 
We call y s the .s th Maclaurin integral limit for y a and 4> (n). If s be negative, the 
previous expansion will hold, but in this case y_ y = oo , and the constant term vanishes. 
Again we have 
m— J rrn 
2 y„ log (n) = | y tl log <f> (n) dn — \ y m log <£ (m) + . . . 
n= l •'■yo 
+ (-y 
B <+1 .4P* 
2t + 2 ! 
y m log <f> (m) + . . 
and 
-1 rm g <+ pt+l 
P>i — | y-n dn ^-/x w + . . . + ( ) 2^ _|_ 2 ! dm it+1 " ‘ 
3 Q 
VOL. CXCIX.—A. 
