422 
MR. E. W. BARNES OX INTEGRAL FUNCTIONS. 
The investigation of the character at infinity of the zero-lines of simple integral 
functions belongs to the theory of functions of real variables. I do not propose to 
undertake it here. It is, however, evident that such lines cannot curl round 
infinity when they belong to functions of finite non-zero order with algebraic zeros :* 
they approach this point in a line which becomes ultimately straight. 
§ 17. A function with a finite number of simple sequences of zeros can evidently 
be built up of a number of functions, each with a single sequence of zeros. 
The function will thus have a finite number of lines of zeros tending to infinity. 
When the zeros of a function of order p are all of the same character and form m 
lines symmetrically ranged round the origin, the function will he equal to a function 
of t (— z m ) of order —. 
Thus a function of order ^ with the sequences 
o-n = n~ 
a,l" — 
where <n 3 = 1, 
is given by the product II 
51=1 
I - w 
r 
which, considered as a function of z 3 , is of order T. 
§ 18. A function, each of whose zeros is repeated a definite number of times, 
h (say), is substantially the k th power of a function with the same sequences of non- 
repeated zeros. 
When the n th zero of a function of simple sequence is repeated a number of times 
dependent upon n, we call the function in brief a simple repeated function. We 
can obviously have repeated functions with any number of sequences of zeros. We 
may, as before, limit our consideration to a function with a single sequence of zeros; 
such a one may be written 
FW = 
n 
51 = 1 
1 A 
5.1 = 1 mV pJ 
The quantity must, in order that the repetition of the zero may not be meaning¬ 
less, be an integral number depending upon n ; but, if we take the principal values 
of the ensuing expressions, it is evident that we may get a generalised repeated 
function by regarding p„ as a general function of n. 
Un 
The quantity p„ must be so chosen that S - r — is convergent. 
5j=l a n P " 
* This statement does not deny that they can curl a finite number of times in the finite part of the 
plane. 
