474 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
Thus, when p — e,p > p, the first term of the expansion of the ratio 
jO O' 
n: p+;-)«-«. 
(~z)P 
V«n' 
n 1 ( 1 + ~lT P ) * «’/P pn plp 
51 = 1 I \ n 
IS 
7T P \ 
sin irp (1 — e x ) 
„0 (l-E)) 
(_ \Pz p ( 6 , 6 n 
And, when p — e x p < p, the first term is -—— 1 Z i p, —]> ] j V ^ ' ' '), which is readily 
( — z)p 00 
seen to he equal to-2 
I> , 1=1 
_ n 
ccj 
n 
p/p 
J' 
§ 69. The expansion which we have obtained is valid for all points except those near 
the line of zeros of the function and for all finite values of the quantities b and e such 
that a n has a finite value. It must he carefully noticed that when any term becomes 
infinite through the occurrence of sines of integral multiples of 77, we must revert to 
the genesis of that term to find the true form of the expansion. Thus, when 
p — pe 1 ~ p, the first term of the ratio just considered is that which arises from 
-hp | + 
, (— ty 1 dt 
=-i- & 1 t 
that is, from 
, ) J.. (— ) s (b 3+ P (m) . . / vl 
- h 'p f - +1 tf^r +< ■-# io §*wi 
p 
where the double accent denotes that the terms corresponding to s = 0 and s = — p 
are to be omitted from the summation. 
<K m ) 
Put now 
the series 
= e l \ then, with the argument previously used, we have to sum 
- b.pfr (m) j j + + (“ <i +ie ) p log <j> (on) j. . 
Now, when p is not integral, we have seen that 
1 (— V 
+ S' e- 
7 r 
V 
s=-l- S + p 
sin it p 
provided — 77 < 6 < 77. 
Let us put p — p + e, where p is a positive integer and e is very small, 
have, retaining only first powers of e, 
1 . J„ ( — ) s c~ si9 , ( — )PcP ie 
_ -j- V" _ 
P s=-k S + p 
+ 
77 
[I + eid], 
so that 
i- (-y e -*u 
— + S v - 
P s=-k S + ]) 
( —)"776 
= ( — )p eP‘ d W, 
Then we 
