380 
,MR. E. \Y. BARNES ON THE THEORY OF THE 
which is the complete asymptotic expansion for log To {z) when 2 ; | is large, and z does 
not lie witliin the Imrrier region negative with respect to the axes of — and — 
If we comhine this result with that obtained in § 83 we find the more general 
formula 
Ion 
& 
To {z + a) 2Si'(o) 
Pi " 2 ) 
= “ 2 ^/ + ^0 Uogy^+„, 2 — ) Trt }+ 2 (a) + ., 81 ®' (a) (y + 
, y 2S',,+ l(«) 
= 1 VI (m + 1) ’ 
valid under tlie assigned limitations. 
The expansion is written in the j)recise form ado})ted, in order that the analogy 
with the corresponding formula in the theory of multiple gamma functions may he 
more clearly displayed. 
§ 85. We might now conclude this investigation. 8 ince, however, this Avoidd appear 
to l)e the first time in analysis in which an asymptotic expansion with a barrier 
I'egion has been obtained, it seems better to give an alternative proof which shall not 
need the difficult argument of §§ 80-82. This proof is the direct extension of that 
})reviously given for tlie case of the simple gamma function."'" We therefore proceed 
as l)riefly as possilde. 
In the investigation of § 57 it was shown that when | 5 | is finite and (•'’) > — k, 
where k is a positive integer, the series for {s, o | wj, m.,) is absolutely convergent. 
8 up 2 )Ose now that s = cr + It, where a > — k, and suppose further that 2 does not 
lie within the region hounded by axes to — and — wo, and that a is positive with 
respect to the w’s. 
Then, since 
(1 — s) (2 — .s) oj] co.i (P 
_I dh _j_ ^ 
2(1 — s) Wo ,, = 0 1 
+1 
■•i + r ’ 
it is evident that, if p lie any positive integer, the absolute value of each term of the 
expression 
ST if 
sin TVS 
28-s, k I Wo) 
tends to zero as lr| tends to infinity. For, liy the restrictions on 2 and a, 
A.-'f 
Y == + g where o < xp < ±: v, 
« (( 
and therefore 
* “ Theory of the Gcamma Function,” Part IV. I regret to say that the Lemma of § 40 is fault}'; the 
theorem is evidently only true when a/w is real. A slight modification will, however, establish the truth 
of the main proposition under the conditions enunciated. 
