480 
MR. E. YT. BARNES ON INTEGRAL FUNCTIONS. 
When pe Y is an integer, we see on evaluating the limit which arises, that the 
dominant term of the asymptotic expansion is still the one just written down. For, 
in this case, the only other term which might he considered first in the asymptotic 
expansion of the quotient is ( — ) i>+l ph 1 z v ~ €ll> log £, which, since is positive, is of 
lower order than 
( — ) p v ry( 6 ,, 
z? Z p, p ; 
P \ b V °2> 
§ 74. It is now evident that, if we are given any simple function of finite integral 
order, we can find its asymptotic expansion. The analysis just given solves 
completely the case of algebraical zeros. When the zeros are not algebraic we may, 
and, in fact, we shall have to introduce new analytical functions defined as indefinite 
integrals ; hut there will be no essential difference in the theory. 
It should be noticed that just as we have to take the principal values of the 
algebraically many-valued expressions which occur in the asymptotic approximation 
for functions of non-integral order, so we must assign principal values to the 
logarithms which occur when the functions are of integral order. 
Part IV. 
The Asymptotic Expansion of Repeated Integral Functions. 
§ 75. As has been stated in the general classification of Part I., an integral 
function, which is such that its n th zero is repeated a number of times dependent 
upon 7i, is called a repeated function. 
If the number of sequences of zeros be not infinite, the function is called a simple 
repeated function; and it is obvious that such a function may be built up of 
functions, each of which possesses a single sequence of zeros. We shall limit 
ourselves to the consideration of such functions. The order of simple repeated 
functions with a single sequence of zeros has been previously defined. Taking this 
definition, we consider, in turn, in the ensuing paragraphs, functions 
(1) of finite (non-zero or zero) order less than unity, 
(2) of finite non-integral order greater than unity, 
(3) of finite integral order greater than or equal to unity. 
And, finally, an example is given of the asymptotic expansion of a repeated 
function with a transcendental index. 
Inasmuch as the principles which underlie the analysis are exactly the same as 
those which have been previously discussed, we shall give but a bare outline of the 
methods by which the results are obtained. 
