ME. E. W. BARNES ON INTEGRAL FUNCTIONS. 
459 
The series for loo’ 
V p (z)*(%Trf 
[ 
exp 
fp t 
7 r 
can be “summed” by the methods of Part IT. 
sm 7 T/p 
The function just written tends to zero near its essential singularity z = co , and the 
same will be true of the function which we get by any process of summation. But, 
in general, the function derived from 
»(-r 1 
■S= 1 
sz 
F (ps) 
w 
. ill not be equal to the function from which the series has been obtained. 
co 
Since F (ps) = £ (—ps), the series is equal to N 
(-) s 
-i 
r (i + s P ) 
s=i s Air 
the integral being taken round the fundamental contour of § 24, 
clx 
(- A 
,p8-l 
(lx, 
The series is thus equal to IG (x p z) — , where Cf (z) = N T (ps) z\ 
The series for G (z) is divergent and of order p. The integral is interesting in that, 
in place of e~ x , we have used (e x — l) _1 as our auxiliary of summation. 
§ 54. We now pass on to consider the most general simple integral function with a 
single sequence of non-repeated zeros, whose order is any number (zero included) less 
than unity. 
1 
The function may be written F (z) = TI 
n = 1 
1 + 
(n)_ 
, where N i . , x t is absolutely 
»=i \H n )\ J 
convergent. The n th zero, — cf) (n), is a definite function of n and any Unite number of 
given constant quantities. 
Suppose that if r = (j) (n), then inversely n = xjj (r). 
Let \z\ = R and suppose that m is a large integer such that m — 1 < \)j (P) < on. 
> 1=1 
<£ (ri) 
t % ~ / Z \ i / 
Then log F ( 2 ) = % log (1 + yqry) = S log (1 + 
0 (n) 
+ 2 log ( 1 + 
<K")/ 
so that if we expand the logarithms in convergent series we shall get 
ra — 1 
log F (z) = (on — l) log 2 — S log (f> ( 01 ) 
n=i 
m — 1 
+ 2 
•a = 1 
'</> (n ) c fr (n) 
2 7 ? 
+ ■ • • + 
(-y ( n ) 
+ ... 
+ 
y 
n=m 
( —y _l z 3 
+ • • • + i -T77TT + 
jt> (n) '2(f) 2 (n) ‘ ' ’ ’ ' $(f s (v) 
Now, by the Maclaurin sum formula, if s be positive 
N _ ft (n) - | ft (n) dn - ±ft (on) + yj ft (in) ...+(-)'- 
m — 1 
t 
«= 1 
B 
/+! 
d 2l+l 
• 
2 t + 2ldm~‘+ 2 
ft (m) -f .. . 
where y is a constant quantity, depending on 5 and the form of r/> ( 01 ), which we have 
proposed to call the Maclaurin integral limit for ft ( 01 ). 
3 N 2 
