MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
423 
We must then, in general, have \J 
n = x> 
1 
Pa 
\_a n n 
= 0 
On 1 
that is to say, a n n p,~ n must 
increase indefinitely with n. We can no longer assign log »asa value for p n , which 
is always sufficient to ensure convergence, as was the case with simple non-repeated 
functions. 
§ 19. It is evident that we may regard the value of p for which, e being a small 
real positive quantity, 
% — p+e is convergent and $ — p _ e is divergent 
7i=l C^ii n=\ etyi 
as the order of the repeated function. When p is an integer, the order is equal to 
the genre : in other cases the genre is the integer next greater than p. 
If the order is not infinite, and the sequence of zeros to a first approximation 
algebraic, p„ must be algebraic also. 
Suppose that 
Lt -1 = 1 and L t ~ 
n r 
/ = co % 
then, we shall have for the determination of p n , pp n — cr > 1, or p /t > (cr -f- l)/p. 
Repeated functions of infinite order will not be considered in the present memoir. 
§ 20. Hitherto we have limited ourselves to integral functions which possess a 
finite number of simple sequences of zeros. But we have not thus exhausted the 
category of integral functions. Instead of the typical zero being a definite function 
of the single number necessary to define its position in the series to which it belongs, 
it may be a function of two or more numbers and belong to a doubly or multiply 
infinite sequence. In such case we say that the function is a double or multiple 
integral function. 
Thus the Weierstrassian cr function is a double integral function, and another 
function of the same category is the double-gamma function to which reference has 
been made in § 3. 
The multiple integral functions always have ultimately a lacunary space* in the 
region near infinity. In the case of Weierstrass’ cr function, this lacunary space 
covers the whole region near infinity ; for the double-gamma function this space lies 
between the negative directions of the axes of w 1 and oj 2 . 
By a well-known theorem due to Jacobi,! functions of treble or higher sequence 
with periodic zeros cannot exist. This theorem may be extended, and we may prove 
that there must, in functions whose sequence is greater than double, be such relations 
* The zeros will, of course, only crowd together indefinitely on the equivalent Neumann sphere. The 
possibility, or otherwise, of summable divergent expansions is the reason for the nomenclature. 
T ‘ Ges. Werke,’ vol. 2, pp. 27-32. 
