DOUBLE GAMIMA FUNCTION. 
2'J5 
In tins expression their principal values are tlironghout to be assigned to the 
logarithms, and the numbers h are to be such that 
= 0, unless (op -f wo) has within the region hounded by axes to — 1 and — in 
Avhich case 
i 1, upper or lower sign being taken as T(w^) is positive or negative, 
while 
= 0, unless Wj lies within the region l)onnded by axes to — 1 and — Wo, in Avhich 
case 
/r, = dz 1, the iip^jer or lower sign being taken as is positive or negative. 
On reduction we now see that 
Fq ^ {Z + OJ) ) 
rrhOOGja),) 
= exp. 
Lt 
(X) 
n = '-0 
{2)1 -|- 1 ) Ct).-) — co^ 
^ - o d“ d~ i) ' d" "i) 
0 0 1 - 
, ( 2// + 1) &)j — &)o I n / . 
lou’ noj., — -z-nig uco^ — log pj(ai0 
2 Wo 
+ p + + - i)o/,._ (1 + 
0).^ 
(0.^ 
cases in which /p and /l'o do not liotli vanish. 
(l) Firstly, w'hen 
OJ 
j does 
("i + Wo) does not 
axes to — i and — Wo. 
/.p = 0 
— i 1 
lie within the region hounded by the 
In this case 
tlie ipiper or lower sign being taken as I(wo) is positive 
or negative. 
And we have 
Fq + mG 
Fj ' (*) Fh^ I w.o) 
= exp. 
71 = » 
S' s' 
0 0 
1 
n 
± 
{'2n + 1) Wo — Wi 1 
---^- log noj.o 
2 w|Wo "" 
•2 + 1)^^ I 2 f '' - 
Wo J \ Wo 
+ (a "F |-) ^ “log a(crj^ + Wo) 
Wj^Wo 
{ 2 ) 1 / -f- 1)W] — Wo 
iWo 
^ log nwj 
the upper or lower sign being taken as I(w.,) is jiositive or negative. 
But in this case m di 1 , the signs Ijeing chosen in the same way. 
If then we take 
/ \ A/ 1 / I 1 \ ^1 + / 1 \ + t) ^3 "" 1 
yoo (wi, Wo) — 2 S — (a “F j) ^ log a(wj^ -F wo) -F log awo 
0 0 
WiWo 
oW.Wo 
. (271 + 1) w, — Wo o , (/i+Dtti 
+-" log nw, ± 2--- 
OJ.-) 
2wo 
