1550251515-Classical_Complex_Analysis__Gonzalez_

622 Chapter^8

We have

roo r t n

lo e-ttz-i dt - lo ( 1 --;i) tz-i dt

=in [e-t - (1- ~) n] tz-i dt + 100 e-ttz-i dt

Given e > 0, there exists Ni such that n ~ Ni implies that

1100 e-te-i dt' < %f

since the r-integral converges when Rez > 0. On the other hand, for

0 :5 t ::; n, or 0 :5 t/n :5 1, we have

t t -i

1 + ;:;: :5 et/n :5 ( 1 - ;:;: )

so that

(

t )-n t ) n

1 + ;; ~ e-t ~ ( 1 - ;:;:

and hence

0:5e -t - ( 1--;:;: t ) n =e -t [ 1-e t ( 1-;:;: t ) n]

Since

we get

e-t - ( 1 - ~ r :5 e-t [ 1 - ( 1 - :: ) n] :5 e-t ~

Therefore, if we let x = Re z, we have

11n [ e-t - ( 1 - ~ r] tz-i dt' :5 ~ 1n e-ttx+l dt

< -^11 e-ttx+i dt = - < -€

00

A 1

n 0 n 2

for n > [2A/e] + l = N 2 , where A = f 000 e-ttx+i dt = r(x + 2). Thus, if

n > max(Ni,N2), we have

1100 e-ttz-i dt - g(n, z)I < f

or