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).