1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions

since

I

1

( :: ~ ) I = 4lx^2 ~ y^2 I

and Ju-v = lx-yl

Finally,· letting u = v + t^2 in the last integral, we obtain

22z-1r(z)r(z + 1/2) = 2100 e-vv2z-1 dv loo e-t2 dt

621

= ..)7rr(2z) (8.20-18)

This formula is due to Legendre .and is called the duplication formula for

the r-function.

Although the formula has been established with the restriction Re z > 0,

by the identity principle for analytic functions the formula holds good for

all values of z for which both sides of the equation are defined, namely, for

all z E C - {O, -%, -1, -^3 / 2 , ••• }.

(11) Again, in view of the identity principle, to prove this property it

suffices to show that

{oo e-ttz-1 dt = lim n!nz

lo n->oo z(z + 1) · · · (z + n)

whenever Rez > 0. To do this, consider the function

r t n

g(n,z)= lo (1-;;:) tz-^1 dt

= nz 11 (1-rtrz-1 dr

where the second integral is obtained from the first by the substitution

t = nr. If Re z > 0, integration by parts gives

rl (1 -T )nrz-l dr = ~ r1 (1 -T r-lTz dr

lo z lo

= n(n - 1) · · ·^2 ·^1 1 Tz n dr

1

+ -1

z(z+l)···(z+n-l) 0

n!

= ~------

z ( z + 1) · · · ( z + n)

Hence

nlnz

g(n,z) = ----· ---

. z(z + 1) · · · (z + n)

and we must show that J 000 e-ttz-l dt = limn_, 00 g(n, z).