Sequences, Series, and Special FunctionsHenceorSincewe obtainorr(x) r(x)
(x+e)x <B(x,e)< (e-1)x
ex exB(x,e) ex
( x + e)x < r( x) < ( e - 1 )xlim ex = lim ex = 1
e-+oo (x + e)X e-+oo (e - l)Xlim ex B(x, e) = 1
e-+oo r(x)B(x,e),..., Cxr(x) as e ---+ 00
(9) From property (4) with z = x > 0, ( = e > 0 we have
B(x, e) = B(x, e + n)B(e, n)
B(x+e,n)
629This formula show that the right-hand side is independent of n. Hence,
taking limits as n ---+ oo and using property (8), we find thatB(x,e)= lim B(x,e+n)B(Cn)
n-+oo B(x+e,n)
= lim (e + n)-xr(x)n-er(e)
n-+oo n-(x+e)r(x + e)
r(x)r(e)
= --'--'---'---'-
r(x + e)(8.20-32)which proves that property (9) holds for z = x > o, ( = e > 0.
However, since B(z, ()is analytic in Rez > 0 for each (with Re ( > 0,
as well as analytic in Re( > 0 for each z with Rez > 0, and f(z) is
also analytic in the half-plane Re z > 0, the identity principle for analytic
functions shows that the formula
B( () = r(z)r(()
z, r(z + () (8.20-33)
holds good for Re z > 0 and Re ( > 0.
Moreover, since the definition of the f-function has been extended to the