DOUBLE GAMMA FUNCTION. 
871) 
= [cSo(2) — .S,|(2 + rt)] [logs; — 2 {m + m')TTC\ 
1 ToCO 
log s; having its principal value with respect to the axis of — ("i + Mr,). 
§ 84 . We may now obtain the asymptotic expansion for log To (2 A- u), under tlie 
limitations assigned at the commencement of the preceding paragraph. 
For this purpose integrate the relation just obtained with respect to a between 
the limits 0 and wj. 
Then, by the formuige of §§ 12 and 76 , we find 
— 5;log^-^^ + M^[logp.{M^, M.^ — (? 71 + ni') 27 noS/(o)] 
+ (1 + 2»^7rt)|Si(2| Wo)+ ~ l<^g 
- [aSi(2) - oSi(2 + M^) + M^ 2^/(^)] [log 2 - 2(«1 + m) 7 n] 
I -,r a I \ O /.A O '/.Xl I ? ( )'"[2Sm 
+ 2[.bo(2 + "1) - obo(2) - „bo (2)] + 2 --• 
))l= 1 
Substitute now the asymptotic expansions for log |Qg UoF -r | 
of which the former has been quoted in the preceding paragraph, and the latter 
obtained in § 79 . 
Then we find 
r2(2| <Ui, Wo) 
— h 
jOo(Wi Wo) 
— — (1 'I'niTTl) 
^()Jl4'>/i')27rt .^{{0 I wi, w.j) 
^1(2!";;)+ 
Q / I \ I ^3 I *^ 3 ) C! 'I \ 
{log 
^a>j + W2 
2('//i + m')iTL] 
+ ^ {^0 I I ";?) ~ ■ 2 So'( 2 ) } + 2 [ — i) log„, 2 — — + 
Wo 
Wo 
+ 
i-Y 
,1=1 {n + 1)2" 
+ ~ <^3loga,, 2-^ .02“- 1~2 r + 
:Wo 
32- 
4 Wo 
I Y ( )” ^ ihrt l o(wo) ^ ,_y, l b,i.+ o(fa>o) W| qS m + |(o) 
,iti .^Zi n{)i f 1) 2" 
Bemember that log„, s: = log„j+„, 2 — 'Ziuttl ; then we obtain l)y an easy reduction, the 
asymptotic expansion 
To (2) e ”0 2S1' (0) 
log -;-s- 
po(Wp Wo) 
= — 3 S 1 ' (2) {log„^ + ,^2 - 2 (m + ni) 7tl] + s; ..S/"' (0) + gS^O) (0) (j + l) 
+ 
“ (-y%sh,+i(Q) 
iZi {'>11' + 1 ) 2 ™ ’ 
3 c 2 
/ 
