1550251515-Classical_Complex_Analysis__Gonzalez_

614 Chapter^8

This integral clearly diverges for Re z :::; 0.

Suppose that z lies in a compact set ]{ contained in the half-plane

Re z > 0. Then there is a constant 8 > 0 such that Re z > 8 for all

z E I<, so

le-1/r 7 -z-11 ::=; 7 -8-1

for r ~ 1. Since ft' dr /r6+^1 converges, the integral (8.20-3) converges
absolutely and uniformly for z E I<.
Next consider the integral

Q(z) =loo e-ttz-1 dt (8.20-4)

and suppose that z lies in an arbitrary compact set I<' C C. Then there

is a constant a such that Re z :::; a for all z E I<', so

w-11 = tRe z-1 :::; tDl-1

for t ~ 1. Since e-t/^2 t01-l -7 0 as t -7 oo, there exists a positive constant

(3, depending on a, such that t01-l :::; (Jet/^2 when t ~ 1. Therefore,

le-ttz-11 ::=; (Je-t/2

and since ft (Je-t/^2 dt converges, it follows that the integral (8.20-4)

converges absolutely and uniformly for z E K'.

Now each z in Re z > 0 can be enclosed in a small closed disk (a compact

set) contained in this half-plane. Hence the integral

loo e-ttz-1 dt

converges absolutely for every z such that Re z > O, so it defines over

this region a function of z called the r-function for short. In view of

Theorem 8.4, we have that the function P( z) is analytic over the region

Rez > 0 while the function Q(z) is analytic in C, i.e., Q(z) is an entire

function. Hence

r(z) = P(z) + Q(z)

is analytic throughout Re z > O, and, according to the same theorem, we

have