DOUBLE GAMMA FUNCTION. 
345 
Putting k = CO, and emplo}dng the asymptotic expansion just obtained for 
oS_^,oo {ct +1} <^i] ''^"6 obtain (.s', a \ oj^, oj^) from the same asymptotic equality 
as that by which it has previously been defined in §§39 and 41. 
§ 59. As an example of the way in which we should proceed in a theory based 
on the double gamma function as defined in § 57, we will prove the relation 
= — \p.2 Wj, (o.i) “h '2{m fi- v/f)7rt.;,Sy (o) 
established in § 55 for the case when a is positive with respect to the w’s. 
In the first place, when 6' = 1 + e and e is small, we see that 
U 
Ci{s, a I (jj^, Wo) + ^ 2So'(a) 
I ^ 3 ) - 2 ^-l-e,o 
1 , O), + Wo 
— c(l — e)w|wy(^“^ 2ea)^a)o 
— - + log a -+ higher powers of e. 
Therefore taking [s, a | w^, wo) as the limit, when ii is infinite, of the sum obtained 
by taking the first (pn fi- 1, -j- 1) terms of the double series 
^ 3(1 + 
N N - — - 
///ii=o «i.>=o c + in^oyy + ^1^2 
— oSQ'[a + {p>>' + l)^^i + + f)^ 2 ] log+ {p‘^ + 1)^] + ( 7 '^ + 1 )^ 2 ] 
+ oS(j'[« + (pn + l)wj]log[a + + l)wi] 
+ + [qn + 1) Wo] log [a fi- (qn + 1) w^] > 
the logarithms having their principal values with res^ject to the axis of — (^1 + Wo). 
Putting a ■= 0, p = <1 =1, we find 
U 
n-O 
.< = 1 
^ 2 (s, a I W^, Wo) + 2 ^(/(o) — y 
— V V' 
Wj + Wo 
0 ^/i^Wj + '«nwo 
4" 1 
-WjWo 
n + 1 
1 
log n — {n-\- 1) (“ + "7 log + Wo) 
\Wl Wo,' 
+ -^'log w^ + log w.o + _p _ iQg / 
Wo (0^ ~ ^Wj^Wo ' ~ ' 
/ \ Wj + Wo . 
= 722(^1, Wo)-T- 2 Tn{m + m ). 
WjWo 
And now, in the limit where n is infinite, we find 
VOL. CXCVI.-A. 2 Y 
