454 
ME, E. W. BARNES ON INTEGRAL FUNCTIONS. 
consider the function F ( 2 ) = IT 
-,i=i 
1 + u 
ir 
m — 1 
, which is known to be equal to 
A 
(27T)" 1 2~*{<W - }. 
We have log F ( 2 ) = E log (l f a) + - log (1 + ~ ), and, if m > \z > m — 1, 
we obtain on expanding the logarithms 
m —1 
log F ( z ) = (m — 1) log z — 2 % log n + 
71=1 
m—1 
V 
-,i=l 
7t~ 
+ ( ^A“ + 
•s: 
+ 
V 
n=m 
z (_y-i 3* 
a - ... + Hat + • • - 
76" 
STfc 2 * 
If now we employ the arithmetic asymptotic approximations given by the 
1 
n~ 
m- 1 
Maclaurin sum formula for log \(m — 1)1}, E -y, and E n~ s , we obtain, in the limit 
v. rn 
when Jc is infinite, 
',i=i 
(~) r 1 B, 
log F (z) = (m - 1) log 2 - 2 (771 - £) log m - m + log ^/2 tt + E l 
„ y-i r ??r j + 1 
+ 2 E 
* ( —y-1 f m~ s+1 m~ s , “ ( — dr 1 ' 
+ 
,= 1 8# [2-S+1 
4. (-rv r i 
•=1 at < 
dm' 2 ’ 
in 
Is 
_ V 
» (— ) r_1 B, tf 2 ' 
S=1 s |_(2s — 1) rn 2s 1 2 m 2s r=1 27’! dm 2 ’ 
m 
or 
log F ( 2 ) = (in — 1) log 2 — (2 in — 1) log in -f- 2 m — log 2 77 
+ 2 (-r 1 
8 = 1 
m if+! 
._*_i + v t-y \ m ~ 
s(2s + 1 ) 2 * s(2-s — 1) m' 2s J J S=1 2 s 
+ s idn* 2 2F£2_! * 
,.=1 (2r)! 1 ^ 
( _y-l rfSr * 
_ --- y — m~ s — E ---777,' r 
S=1 ss* diri- T s= i s dm~ r 
RrAAl,„-4 
where we have re-arranged the terms of our double series in accordance with § 43. 
Now by the theory of summable divergent series 
Til" 
= - log 
in- 
and 
4 (- y r« a 
*=1 S 1 2* 
. ou_ro; 
— 2 - 0 - ' -f E ---- 771~’ f = 0. 
m 2r-l I J-2- 
*=-* dm- r 
Hence we have, when ??i is large, the approximation, asymptotic with regard to in, 
log F ( 2 ) = in log -A + 2 ,in — log (2772*) 
+ 22 (-WA b 
2,' ' I ' - s 2s + 2 2 ? T Ws - 1 2s ) M-'-' | 
