628 Chapters
or,
B(z,( + 1) = ~B(z,()
z+.,,if, in addition, z + ( f= 0.
(4) For n = 1 we have, using properties 1 and 3, with the indicated
restrictions,
B(z, ( + l)B((, 1) = [(/(z + ()]B(z, ()(1/() = B(z ()
B(z+(,1) 1/(z+() '
If the property holds for n 2: 1 it also holds for n + 1, since
B(z, ( + n + l)B((, n + 1).B(z+(,n+l)
= [((+n)/(z+(+n)]B(z,(+n)[n/((+n)]B((,n) =B(z,()
[n/(z + ( + n)]B(z + (, n)Thus, by mathematical induction, property 4 holds with the restrictions
Rez > 0, Re(> -n, Re(z + () > 0, ( f= 0, n 2: 1 an arbitrary integer.
(5) By using properties 2 and 3, we obtain
B(z+l,()+B(z,(+1)= ~B(z,()+ ~B(z,()=B(z,()
z+.,, z+.,,provided that Rez > -1, Re(> -1, z f= 0, ( f= 0, z + ( f= 0.
(6) Again, by using (2) and (3),z.B(z, ( + 1) = z( ;-B(z, () = (.B(z + 1, ()
z+.,,with the same restrictions.
(7) The substitution t = sin^2 (} transforms the B-integral into
r-/2B( z, () = 2 Jo sin^2 z-l (} cos^2 (-l (} d(}
provided that Re z > 0, Re ( > 0.
(8.20-31)(8) Since et 2: 1 + t fort real, with equality holding fort = 0 only, we
have for x > 0, e > o, and using (8.20-14) and (8.20-8),
I'(x) = {°" e-(xH)ttx-1 dt < {oo tx-1 dt = B(x e)
(x+e)"' Jo Jo (l+t)"'H '
and for~ > 1,
___!:.(:l_ = f
00
e-<e-l)ttx-l dt > f1
t"'-^1 (1-t)e-^1 dt = B(x e)
ce--1)"' lo lo '