MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
473 
lil — 1 
m -1 
2 a/ = % 
n= 1 
91= 1 
HP + sb^'iP 1 + 
e l I S (s I) i o — — 2ej 
b^np -j- sboUP + . . 
/ o \ 
= j c f) s (n)dn + F (—) + sb r F 
+ 5 ^sF 
P. 
e 2 ) + • • • 
S 
\ P 
e l + 
s(s-l) 
b>F - - 
- p 
*<*>+ I 
^ ,f r 21+21 An“+' v ' ’ 
9 
- e l , s (s + ] ) , „ - i - 2e, 
— p 
7/1-1 ^ 7,1-1 r _ S 
and X -- = X n ? — sAn / 1 + 
*=i«/ 77=1 L 1 
= fV(»)rf» + z( P> -»; ££;;;) + f(- p - ^ 
s ai,i!“ ,,,, 
i i 0 LU 4 2! ^ W* 
where Z i p, — s; ^ ^ ' J caii be expressed in terms of Riemann £ functions, or the 
equivalent Maclaurin constants F by the formula 
Z [P> ~ S ; b\ b, ’ ’ '.) ~ “ i96iF (~ p ~ e i) + 
s(.s‘+ 1) 
&! 2 F 
- 2e n 
*&,F - ~ - e 3 + 
As formerly, we put 
z(o : £ £;;;) = lf (-«,)-£*(- 2 *)+w-%)+•••. 
so that log 2 -tt + Z(0) arises as the Maclaurin constant corresponding to the 
7)7-1 
asymptotic expansion for X log <{> (n). 
77 = 1 
Proceeding exactly as for the case when the order of the function is equal to unity 
we see that the asymptotic expansion of log P (z) is 
7 r 
00 (— V 1 / <5 
-- z p — A log z — - log 27r + X' - F 
sin 7 rp ^ & 2 p ^ s= _ p sA \ p 
TT 
sin7r/j(l — ej) 
— 7Tp& 2 
gPd-O 4 . dlu + i - 2 PU) h 2 
77 
sill 7rp (1 — 2e L ) 
..p(l- 2 e,) 
• p (1-el 
o 7 _ y-i 
r?/ V / 13 V 
, x ~ + . . . + 2 
Sin 7rp (1 — 6 2 ) s =-j, 52 
p, S 
€], fo ■ • . 
; & 1; 6, . - . 
this exp<ansion being valid when — 77 < arg 2 < 77. 
3 p 
VOL. cxcix.—A. 
