MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
42] 
infinite order. On the other hand, if oq increases faster than any algebraic power of 
n, however large, provided it be not actually infinite, the function is of zero order. 
In other words, functions whose order is “ finite both ways,” to use De Morgan’s 
phrase, have zeros which are to a first approximation algebraic. 
The zeros of the function are said to be actually algebraic when they are given by 
o n = c 0 n 
7 P 
1 + + - + 
n pi n Pi 
when p is of course positive and rational, the c’s are constants, and p x , p . . . are 
in ascending order of magnitude. 
It is now evident that we can form a scale of integral functions; thus, in between 
functions with the algebraic zeros 
a„ = n Pl and a n = n Pl , where p. : > p { , 
will come functions with zeros like 
n Pl log ??, n p> log n . log log n and so on.* 
Such functions we call simple integral functions of finite order with a single 
simple transcendental sequence of zeros; or, in brief, functions of transcendental 
sequence. 
Tims 
n 
71=1 
1 + 
(n log ?i) 3 J 
is a function of transcendental sequence of order \ and genre zero. 
§ 15. Functions of zero order, which must always be of transcendental sequence, 
co 
can be classified in the same way. The most simple is II 
n=l 
1 + 
c n 
Then we consider functions whose zeros are obtained by multiplying c n by an 
algebraic function of n. The next step is obviously to introduce intermediate 
functions by means of logarithmic terms, and so on. Then we introduce functions 
formed from sets of zeros of still more emphatic convergence, such as 
n 
n=i 
The range is obviously limitless. 
§ 16. It is worth noticing that the density of the zeros along the (possibly curved) 
line on which they lie, decreases with the increase of the convergence of the function. 
The zeros of the higher functions of zero order have therefore a density which 
becomes less as we go higher. The conception of the density of a function is perhaps 
the most easy way of intuitively classifying it. 
* The analogy of the De Morgan and Bertrand scales of convergence is almost too obvious to need 
mention. 
