1550251515-Classical_Complex_Analysis__Gonzalez_

604 Chapter^8

4. Find the Laurent expansion of
   1
   f(z) = (z - l)(z - 2)(z - 3)

in each of the following regions: (a) 1 < lzl < 2; (b) 2 < lzl < 3.

5. Show that

sinz = ~ + ~ (-l)n z2n-1

z^2 z ~ (2n + 1 )!

for 0 < lzl < oo.

6. Show that

cscz= ~ + ~ 1 z-[:, -( 3 ~) 2 ] z

3

+···

for 0 < lzl < tr.

7. If f(z) is analytic for lzl < 2.5, show that the composite function

f(z+z-^1 ) has a Laurent expansion in powers of z valid for% < lzl < 2.

8. Prove that the Laurent expansion in powers of z of f(z) = sin t(z+z-^1 ),

valid for lzl > O, is of the form

where

Fig. 8.11

co

Ao+ L An(zn + z-n)

n=l

1r

An = ~ J cos nB sin(2t cos B) dB

0