470 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
log Q 0 («) = (m - 1) log z— 1 (m - 1) log to - — - - 1 log 2tt 
P P ~P 
+ I' F g' 
l p 
s=-p 
+ 
mp 3'J p\i (-) ~ UA 1 (-) 
, £ B/+i f - (2t )! i, (—) 3_1 
^ Zo 2t + 2\ 1 P S 
\z] 
'cl 2l+l p" 
d, 2 '^ 
r=\ 
\Z\ . . 
We suppose now that ^- l p is a quantity which, when m is very large, is ultimately 
equal to unity. Then we may “ sum,” as before (§ 52), the Fourier series, which are 
the various coefficients in the preceding expansion. 
ttt *. (—)® 1 fm l lp\ s 7 t z° . m 1 p 1 . , . 
We have p 2 - —- — = 4- Log- — —, provided p is not integral, 
r ■ o - -L o '■ 2 / Sill 7 Tp m * z p 1 r ’ 
*=—t s. p + s \ 
^ f —V 1 ! n\}\ \ s 
and provided — 7r < 0 < tt. And — tt 2 f —— j = — ^ log 
Also, exactly as before, the coefficient of B^ + 1 in the asymptotic approximation 
for log Q p ( 2 ) vanishes identically. 
Therefore we have, provided — tt < 6 < 77 , the asymptotic equality 
l°g Q P ( 2 ) = ( m ~ i) log - m - 1 log 2 - log 277 + 2' F ( s ) 
-p S= —p s ~ \ P / 
7TZ P , . Vp'P m 1 
+ --4- to log — — — 4 log- 
1 Sill 7775 & 2 p - * z 
Thus, provided — tt < arg 2 < 77 , we have finally 
* Q. <*>=- i lo § * - * >°g *+, a ( -iF F (7) 
This expansion is exactly analogous to the one previously obtained for log P p ( 2 ) 
and is to be regarded in the same way. It must be borne in mind that n l,p has been 
assumed to he the arithmetic p th root of n. Had any other root been taken—say 
27 nr 
the arithmetic root multiplied by cj = e -, where r is an integer, we should have 
obtained the asymptotic expansion 
— (2nr)~* p 
Qp (*) = e p ex P 
7 tz 
L sin it p 
+ 2 
s= —p 
SZ s 
F In¬ 
valid, when — 77 < 6 — r — < 7 r, i.e., when — 7r + < d < 7 r + 
P P P 
The expansion is thus valid everywhere except along the new line of zeros. 
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