616 Chapter^8
with respect to z, and also with respect to (, whenever Re z > 0 and
Re(> 0.
Proof By means of the transformation t = r/(1 + r) we may bring the
given integral into the form
l
oo Tz-1 dr fl Tz-1 dr roo Tz-1 dr
B(z, () = o (1 + T )zH = Jo (1 + r )zH + J1 (1 + T )zH (8.20-8)which is somewhat easier to handle. For the first integral on the right
of (8.20-8) we have, for 0 < r :::; 1 and Re z > 0, Re ( > O,
T _ < 7 Rez-l
Iz-1 I ~ez-1
(1 + r)zH - (1 + r)Rez+Re CSince f 01 rRe z-l dr = 1/ Re z if Re z > 0, we conclude that the first integral
converges absolutely provided that Re z > 0, Re ( > 0.
As to the second integral, we have, for r 2: 1 and Re z > O, Re ( > O,
I
Tz-1 I rRez-1 < TRez-1
(l+r)zH = (l+r)Rez+ReC 7 Rez+ReC
But Jt drjrReC+l = 1/Re( if Re(> 0. Hence the second integral con-
verges absolutely, again with the proviso that Re z > O, Re ( > 0. Clearly,
the first integral converges uniformly for Rez ;?:: 6, Re( > 0, while the
second converges uniformly for Re z > 0, Re ( ;?:: 6 ( 6 > 0). Hence, for each
fixed ((with Re ( > 0), B(z, ()is an analytic function of z in Rez > O, and
similarly, for each fixed z (with Rez > 0), B(z,() is an analytic function
of ( in Re ( > 0, as follows from Theorem 8.4. ·
Theorem 8.48 The r-function has the following properties:
i. r(1) = 1
- r('%) = -.fo
- r(z + 1) = zr(z) (z "" o, -1, -2, ... )
4. r(z + n) = z(z + 1) ... (z + n - l)r(z) (n > 1 an integer)
- r( n + 1) = n!
6. r( x ), x > 0, is convex
- lim.H-n r(z) = oo (n ;?:: 0 an integer)
8. r(z)r(l - z) = 7r/sin7TZ
9. };^00 e-pttz-^1 dt = r(z)/pz, Rep> O,Rez > 0
- 2~z--^1 r(z)r(z + %) = J?il'(2z)