1550251515-Classical_Complex_Analysis__Gonzalez_

600

Now

1

z-2

1 1 oo 2n-l

-; 1 - 2; z = 'E ---;-;;-

n=l

valid for 12/zl < ll, or lzl > 2, and

1 1 1 00 Zn

4 - z = 4 1 - z/4 = L 4n+i

n=O

Chapter 8

(8.18-16)

(8.18-17)

valid for lz/41 < 1, or lzl < 4. Hence combining (8.18-16) and (8.18-17),
we obtain

1 oo 2n-l 1 oo Zn

f(z) = 2 L ---;-;;-+ 2 L 4n+1

n=l n=O

which converges for 2 < lzl < 4.

(b) We_ have

1 1 1 00 zn

z - 2 = - 2 1 - z 12 = - 'E 2n+1

n=O

(8.18-18)

valid for lz/21 < 1, or lzl < 2. Hence combining (8.18-17) and (8.18-18),
we obtain

1 oo Zn 1 oo Zn oo 1 - 2n+l

J(z) = -2 L 2n+1 + 2 L 4n+i = L 22n+3 Zn

n=O n=O n=O

which converges for lzJ < 2.
(c) We have
1 1 1 oo 4n-l
4 - z .= --; 1-4/z = -'E ---;-;;-
n=l

(8.18-19)

valid for 14/zl < 1, or JzJ > 4. Hence combining (8.18-16) and (8.18-19),

we get

which converges for lzl > 4.

We wish to check the last result by applying formula (8.18-2) so as to
illustrate its use in this particular case. We have

Am=-1/1(1 - --+--1) --dt^1

27ri 2 t - 2 4 - t tm+l

r