452 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
Part III. 
The Asymptotic Expansion of Simple Integral Functions. 
§ 47 . We now proceed to consider in detail simple integral functions. After the 
discussion given in Part I., we may confine ourselves to functions with a single 
sequence of zeros. 
We shall find that such functions divide themselves naturally into three groups :— 
(1) Functions whose order is less than unity, 
(2) Functions of non-integral order greater than unity, 
(3) Functions of integral order greater than unity. 
In connection with each group of functions with algebraic sequence of zeros we 
first consider a standard tyj3e with which all functions of the group may be 
compared. 
These standard functions are 
P,(4= n 
n =1 
1 + 
, where p > 1. 
Q, 0 ) = n 
1 + .Mr e 
o — ..I/p +■■■+' 
pn 
PIP 
is an integer such that p ~b 1 > p > p. 
( - zf 
P p (Z) = n 
n=l 
1 + -v P Ve-*^ + --- + 
pn 
, where p is > 1 and not integral, and p 
, where p is an integer > 1. 
For the logarithms of each of these functions we obtain in turn the complete 
asymptotic expansion near 2 = oo. We then show how all functions of the same 
order with algebraic sequence of zeros yield by the same method similar asymptotic 
expansions. And we indicate how it is possible to apply the same methods to wide 
classes of simple functions with a transcendental sequence of zeros. 
$ 48 . The constants which enter into the analysis arise from the Maclaurin sum 
formula (§ 41 ), which may for our present purpose be written 
w.-l rm T> r 7 
S '!•■ (n) = | </>’ (n) dn - i </>• (m) + — — 4 >' (m) + 
(2 1 + 2)! dvv Lt+1 
s being any integer, positive or negative. 
What we have called the Maclaurin restrictions for the function <f>(z) are always 
