1. BASIC INTEGRAL REPRESENTATIONS 15
simple poles at s = 1, 2, •, r. It is also legitimate to differentiate arbitrary times
with respect to s under the integral sign. Let i be a positive integer. We obtain
(1.6)
s=0
In particular, we have
(1.7)
(r(0,Ul,x) =
27T.
dz
(1-8) -7^(r(s,V,x)
e-ujkz^
z
'
log(-z) 1 f
e~xz
log(-z) ,
= F = / ^FT?—r, r dz + 7(r(0,u;,x),
since r'(l ) = —7. Here 7 denotes Euler's constant. Differentiating the formula
(1.5) i times with respect to s at 5 = 0, we have
(L9)
- ^ r £
Q(-i)jr«(i)^e—oog(-^)rjf
= (-logxy.
We define
(1.10)
3=0
-\ogpW(w)= lim
(g-)'Cr(5,o;,x) + (-ir+1(logx)
s = 0
By (1.6) and (1.9), we have
(foYCr&UiX) + (-iy+1(io x)
sO'-^'/^^-^c-^'-rcr^-gs=0
Since
i - I E = i d -
') :
i s rapidly decreasing on L for |z| —• 00, the integral
zllfe=i(l-e- w **)
converges for a; —• +0. Hence the limit in (1.10) exists and we obtain
(1.11)
3=0
2TT ^ \ j
1— n
^J
•Ln-'H-gf--^?}*-
(1.12)
1.3. Now we define a generalization of the r-ple gamma function by
lf(x,o;)
f
d
l o g
" [ * ] / / =(^-J^(s^,x)
pV[u) os
s = 0
The r-ple gamma function introduced by Barnes is the case i = 1. In this case, we
put
Tr(x,u) = rW(x,w), Pr(u) = pW(u).
Previous Page Next Page