14 I. MULTIPLE GAMMA FUNCTIO N AND ITS GENERALIZATIONS
and we are going to show
(1.3) J^ (x +
Q)-a SCr(x)x-a+r,
ar
£l=miuJi-\-m2UJ2-\ \-mrujr
by induction on r. Suppose that (1.3) holds. Then, for a r + 1, we have
E (x + ny*
Q,=miui-\-m2UJ2-\ \-mrujr-)-mr+iujr+i
oo
^Cr(x + m
r +
iC;
r +
i ) 22 (
X
+
m
r + l ^ r + l )
_ a + r
using (1.3). Since Cr(x + m
r +
ia;
r +
i) ^ Cr(x), we can majorize the series on the
right-hand side by
Cr{x) ( - + 7 ^ T r } X _ a + r + 1 = a
+
l ( ^ ) x ^ + r + 1
[ x u ;
r
+ i ( 7 - r - l ) J
using (1.2). Thus we have proved (1.3) and the assertion on the convergence of
(1.1) follows.
We see that d(s , l,x) =
J2m=o(x
+
rn)~s
coincides with the Hurwitz zeta
function and
Ci(5
1? 1) =
C(5)-
1.2. Let L denote the contour which starts from -foo, runs on the real axis,
encircling the origin once counter-clockwise on the circle of a small radius e with the
center at 0, runs the real axis and returns to +oo. We choose the branch of log(—z)
on L so that it takes real values when z 0. We put
(—z)s~1
= exp((s 1) log(—z)).
Then, if 3?(s) r, we have
f
e~xz
I =^—
7
-(-z)s~1dz
f°° e~xt
= 2 v / = r l s i n ( ( 5 - 1 ) T T ) / = 7^ts~ldt
= - 2 ^ / ^ s i n 7 r s ^ ( s ) ] ^ ( £ +
ft)-s
where ft extends as in (1.1). Here notice the difference of (—z)s~l on LPlR
+
when
z is going or returning way viewed from +oo; also notice that it is permissible to
schrink the small circle if e 2-K JUJ^ for all k. Hence we obtain
-ir(i-s)-f/ . ,
S
-
K
^ Ani=i(i-e-^)'
(1.4) Sr(8,W,x) = 1 J = ^ ~ -—{-zy-Hz.
The contour integral of this type has the origin in Riemann's celebrated paper [Ri].
As a simpler case, we have
(1.5) x~s = v ~lT}1 S) [ (-zy-'e-^dz.
2TT JL
The integral j
L
in (1.4) converges absolutely for all s and the convergence is
locally uniform with respect to s. This is the main virtue to obtain a contour integral
representation for an analytic function. In particular, we see that £r(s,cc;, x) can be
continued meromorphically to the whole s-plane, holomorphic except for possible
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