498 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
^ 1 
+ 
sin 77 -p (1 — e 2 ) 
pip + 1 ~ 2 <h) 
2 
irpb 2 
yP(l- e l) 
7 rp 2 5 i (l — e^ 
sin irp ( — 6j) 
az 
0 (1 — 0 — 1 
4 * • • • 
A Cl 
7T 
? p(l-26!) 
sin irp (1 — 2e x ) 
+ 
p~ (p + 1 — 2et) (1 — 2eJ 
ira 
0(1-260-1 
sin 7r p (1 — 2e x ) 
+ • • • 
- 2 p(1 "^ + S' { -^z(p, s ; ^ ’ 
sill 7rp(l - e 2 ) s= _ p \ r b v o, 2 . . . 
- z ( 0 ; oz^p, 
e n e -2 • • \ , 
p '• h .. 1 + ••• 
By the employment of extended Riemann £ functions of parameter a, it is impossible 
to give a form of this expansion which shall include all powers of a , analogous to the 
expansion of log V (z + a), which involves Bernoullian functions of a as coefficients. 
For brevity we content ourselves with the preceding first approximation. 
§ 96 . By differentiating the expansion for log P (2), given in § 68, we have at 
once, as is evident by the preceding paragraph, 
p/ ( s ) _ irp zP -i _ L 
P (fi) sin 77-p ’ 2 z 
_ VM 1 -O 2 p(1-0-1 
sin 77 "p (1 — e 1 ) 
+ (-)'«F (—- ) + ■ • 
\ P 
TTP~M 1 ~ ^,(1-0-1 
sin tt p (1 — e 2 ) 
+ (-)v-z(p,-p;.^;;_ 
Thus the asymptotic expansion for log P' (2) is given by 
7 T 
Sin 7 Tp 
irp\ 
+ [p — log 2 — log 277 + log — - + S' -— \ — F 
V 2 / ® 2/d s & sm 77 p s= _„ 
sin 77/5 (1 — 6j) 
*P(l - 0 
+ 
-P 
pip + 1 ~ 2 P e i) 
S=-P 
V -■ 
77 
yP (1-20 
sin 77/5 (1 — e a ) 
, 5 / l~) s 1 7 / . . 6 1> e 2 
+ i, v z (^ s ’ 
ftp & 2 • • 
z(°; + terms involving positive 
(fractional) powers of \/z. 
d 
Thus — P (z) is a function of the type 
LIZ 
CO 
/ z \ - 2 + . . 
(-) P * P 1 
n 
n= 1 
[{ 1 + K) e 
] 1 
O 
©1 
where h„ — n 1/p 
b-. b n 
1 + — + — + . 
^ % £l T T 
n 
i/p 
-- T- higher powers of— 
/m 9 r 
Thus the differential of an integral function of order p (> 1 ), where p is not 
integral, is itself an integral function of order p whose n th zero, when n is large, will 
