1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 601

= 4~i [/ (i _ :;tm+i +I (4-:;tm+i]

where r is any circle with center at the origin and radius greater than 4,

described once in the positive direction.

Assuming that m = n :2: 0 and decomposing the integrands into simple

fractions, we have

But

and

Hence

if n :2: 0.

-(t---2--tn_+_l = 2n~l ( i ~ 2 - ~ - i~ - • • • - t::l)

1 -1 ( 1 1 4 4n )

( 4 - t)tn+l = 4n+i t -4 - t - t 2 - ... - tn+l

J

:!! = J :!! = 27l"i

t-2 t-4

r r

J

ckdt= {

0

.

27ri

r

if k ¥- 1

if k = 1

Form = -n < 0, we have

A-n = ~ (! tn-l dt -! tn-1 dt)

47ri t - 2 t - 4

r r

and by Cauchy's formula, we obtain

A_n = l/ 2 (2n-l _ 4n-l) = 2n-2{l _ 2 n-1)

so that

as before.

2. To find the Laurent expansion of f(z) = ez /z^3 valid for any z f-0.

Any given z ¥- 0 can be enclosed in a circular ring with center at the origin

and radii r and R such that r < lzl < R. In such a ring f(z) is analytic