DOUBLE GAMMA FUXCTION. 377 
§82. But when we come to the case of the negative quasi-quadrant given l)y 
it is interesting to notice that the above proof breaks down. 
As Ijefore, 1)y the use of the fundamental diflerence equation, we obtain the 
relation 
log r,,(a — n^o)^ ~ n., 0 }.,) — log lh(o) — log r,i(« — wj — w,,) 
— log lh(n- — 01.1 j o)|, —■ ( 02 ) — log ro(n ■— o)| — co.^l —o)j, 
I\(o — I a)o) 
OJ 
.c) 
log r,i(« — o»j j — o)j, Wo) d- S lo. 
„l, = 1 
+ 
do 
Pi(coo) 
Bto. — VI,,(0.2 I &)[) 
'do '1^0 
s s 
nl.y =1 'y/2 J = 1 
— log ro(u — o).-, I Wo, — Wj) q- 2 log 
-/I,, = 1 
'do 
log ro(« — W| — Wo I — Wj, — Wo) +2 2 log ('< — 7 ?qojj — 
— 2 2/71771 S/(a — I C(J^) — 2 2'77f 77(, S/((f — ')7;oWo|w|) . 
//ij = 1 Vdo =1 _J 
The several expressions bracketed on the right-hand side of this identity admit of 
asymptotic expansions in powers of ^ and —, of which the teiins involve a 
algel)raically ; and therefore the whole of the right-liand side admits of an expansion 
of this form. But there remain the non-algebraic terms 
log' Eo^C^) -|- log — W| I — Wj, Wo) “h log To (ci —■ Wo I Wj, “■ Wo) 
+ log r 3 (« - OJi - Wo 1 - Wj, — Wo), 
and when we seek to group — 7?jWj — 7?oWo with a, we are forced Ijack on the original 
function ro(« — "/qwj — Thus as regards the possibility of an expansion, when 
2 is negative with regard to the w’s, our results are, as we should expect from § 78, 
entirely negative. The region between the axes of — Wj and — wo is a harrier-region 
for the asymptotic expansion of tlie doulde gamma function. When Wj = wo this 
region closes up into the harrier-line which occurs for the G and sinqjle gamma 
functions. 
§ 83. We can uow find the asymptotic expansion of 
1 1 0 {z -f- c I Wi, W.i) 
lOO’ !-i- - 
■ ^ GGhawd ’ 
for large values of | 2 | which are such that 2 does not lie in tlie baiTier-region, a being 
any complex quantity of finite modulus. 
For such values of 2 and a we have the expansion 
VOL CXCVI.-A. 3 C 
