472 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
T1 le final integral gives rise to the dominant term of the asymptotic expansion. 
As formerly, it may be written 
Lt | L\fj {z 
k=x 
sin {k + I>)/4 
sin ^ fji 
dg. 
If we denote the value of this integral by f ( 2 ), and if we put 
f l°g $ ( n ) dn = log F 0> | cfy (n) dn = F„ 
we have the final asymptotic equality 
co 
II 
»i=i 
exp 
/« + *' 
S=-p 
§ 67. We notice the exact analogy between this expansion and the one previously 
obtained in the case when the order of the function is less than unity. The only 
difference arises from the Maclaurin constants. In the former case, all the constants 
corresponding to negative values of s were zero; in the present case, the first p of 
7/i—l J rwi 
them are formed from asymptotic expansions like % — = <f)~ s (n) dn + . . . , and 
n=l &11 J y—s 
give rise consequently to finite constants ; while only the remaining ones, formed 
co ^ r m 
from expansions like — X — = (71) dn + . . . are such that y_ s = co . 
n=i)i Q- n •'' r-s 
We notice also the great elegance with which Weierstrass’ exponential factor 
enters to ensure the finiteness of the expressions obtained in the course of the 
analysis. Could we conceive an attempt to investigate, for functions of order greater 
than unity, the theory which we carried out for functions of order less than unity in 
the first paragraphs of this part of the present paper, we should at the outset be 
forced to invent again Weierstrass’ great theorem. 
Application to Functions with Algebraic Sequence of Zeros. 
§ 68. We will now evaluate the first few terms of the asymptotic expansion for 
CO 
, _ (-)PzP ! 
/ z\ -b - 1 * 
1 + \ + h +. 
= II 
Id -e “* 2 I, where a n — np 
, , 
1 
_\ ««/ 
n 1 n - 
_ 
and the 
e’s are positive real quantities arranged in ascending order of magnitude. 
Let r = a n = (f) (n ), then on reversion of series we find 
n = 1 p(r) = - pbp^ + ,*<!-*>> _ p b 2 f'-^ + . . 
When s is positive, 
