374 
ME. E. W. BAEXES OX THE THEORY OF THE 
+ “ i ‘°s" + r ®(«l -)+ fS 
m = 1 ("*' + 1 )-™ 
with a barrier-line along the axis of — oj. 
When a = w, we see that we have the asymptotic expansion 
IVi±2LlO =/l _ m 3^4 + I 
° Piifo) \13 2a)'/ ^ 4a)“ = i mcoz’^ 
§ 80. We will now jDrove that, provided 2 be positive with respect to the w’s, there 
exists an asymptotic expansion for log Fj ( 2 ) of the form 
( 1 , 2)2 . _j_ + 
V 
A = 1 
where ( 1 , 2)2 denotes a quadratic function of 2 . 
In the first place it is evident that we have 
2 — cc -j- ~h 
where a is some finite quantity, and Uj and ?Zo are singly or together large positive 
integers. 
Now, from the fundamental difference equations of the double gamma function, it is 
at once seen that 
log (a) - log Fo {a + + 
n ”3 / , X , n F, (rt + WiO), lo),) 
= log n n (a -f- m,co, + m.xco.d 4 - log H - ^ —=- 
" m,=0 nu=0 - -/ • o 
+ log H 
3 Fi (a + m„(o^ I a)|) 
77lo = 0 
Pi("i) 
— 2m TTL X S/ {a m^co^ \ oj^) 
7nj=0 
— 2niTTL X S 2 '(a + | Wo). 
?no=0 
A term has been neglected which is an integral multiple of 27tl, and which is there¬ 
fore absorbed by a suitable .specification of the logarithms involved. The above 
formula may be rewritten 
log Fg (« + n^COj -|- « 2 (iJ 2 ) = 
+ 
1 1-1 / \ 1 TT A 4 - ??qa)j I 0)2) 
log I o (a) - log n - ’ Y - 
Jiii = 0 Pi 
1 / \ 1 14 hffc + wpwolo),) 
log To (a) — log IT --- 
)H., = 0 Pi (<^]) 
7?1 ?lo 
log Fo («) + log IT TT (a + 
mj — 0 j >?-2 = 0 
+ ( 1 , ^ 0 ' + ( 1 ) «)". 
In the first place, if we put in place of and n.^ in place of q7i in the procedure 
