1550251515-Classical_Complex_Analysis__Gonzalez_

618 Chapter^8

= (z + n -l)(z + n - 2)r(z + n - 2)

= (z + n - l)(z + n - 2) · · · (z + l)zr(z) (8.20-11)

By writing this formula in the form

r(z) = r(z + n)

z(z + 1) · · · (z + n - 1)

we may define r(z) in the strip -n < Rez:::; -n + 1, z '=/= -n + 1 in terms

of the values of r(z) in 0 < Rez :::; 1.

E I r( 3 5)

_. r(o.s) _ ..!§.... ~

xamp e. - · - (-a.s)(-2.s)(-i.s)(-o.s) - 10s y7r.

(5) By letting z = 1 in (8.20-11), we get

r(n + 1) = n(n -1) ... 2.1r(1) = n! (8.20-12)

Thus r( z + 1) generalizes the factorial function of the· natural numbers.
In fact, it was the problem of generalizing n! to real values that led Euler

to the definition o:f the gamma function. It must be pointed out that this

interpolation problem does not admit a unique solution, for if g(z) is any
meromorphic periodic function of period 1, with g(l) = 1, and we define
f(z) = g(z)r(z), then ·

f(z + n) = g(z + n)r(z + n) = g(z)r(z + n)

and letting z = ), we find that

f(n + 1) = g(l)r(n + 1) = n!

For instance, we may take g(z) = cos27r(z - 1).

Remark Formula (8.20-12) is also valid for n = 0 with the usual
convention O! = 1.

(6) This property follows from the fact that

r"(x) = 1

00

e-^1 t"'-^1 (1nt)^2 dt > 0 for x > 0

0. (7) From (8.20-10) and the analyticity (hence continuity) of r at z = 1,
   we get

1 1m. r( z=1m ) 1. r( z + 1) =oo

z-+ff z-+0 z

Similarly, the formula

r(z)~ r(z+n+l)

z(z + 1) .. · (z + n)