DOUBLE GAMMA FUNCTION. 
290 
V.f 1 {z \ Wo) = c - ^ . z . n U' 
= 0 m .2 = 0 L 
l + ^ie 
" n 2n2 
where and yoo(w^, Wo) are two constants, which Ave call the first and second 
double gamma modular functions of the parameters and wo. 
These constants are given by the relations 
y. 2 i (wj, Wo) = Li 
1 1 
— log«-S_ 
nil = 0 rii2 = 0 
1 
d- {log W^ + log Wo - log (W]| d" Wo)l 
Wif/Jj 
yoo (oj^, Wo) = Li 
^ *^1 T Wo 
. 7 )ll = 0 III., = 0 tt 
A- lo(T n 
2w^wo ^ 
n. + I 
Wo 
log 1 d- - 
Wo 
Wi 
-log (^1 d-+ -T) -(log H" ~ ~ *^ 2 } i ’ 
Oy.-) \ ^3/ — 
AAdrere the logarithms are such that their principal values must ahA'ays l)e taken. 
And the theory is the natural extension of that of the simple gamma function 
r](2:lw^), Avhich is such that 
= 1 
ri-i(2l wj) = . s. n 
m, = 1 
1 +v— e 
where the constant y^ is given by the relation 
r 1 1 
7ii("j) = Li N -- 
)i = » L = U^pWj Wj 
log U w, 
and the principal Aarlue of the logarithm must again be taken. 
§ 25 . We may now see at once that 
Ug (W] I Wj, Wo) — ^y ( 27 r/wo) . 
To (wo } o)^, CO2) — *^1) • ^ 
For we have seen that 
To fiz + W,) rUsIwo) -2m7rd--d 
- — — 1—::— p \oj2 y 
^A 27 r/co,) 
where m is either zero or unity according to the determination of § 22. 
But Li [2r2(2)] = I, 
2 = 0 
and Li [zri(2lwo)] = 1 , 
j= 0 
as we see immediately from tlie product expression for 
