8.1 • THE RESIDUE THEOREM 295- EXAMPLE 8 .4 Find the residue off (z) = " <^0 ;<"•) at zo = O.
Solution We write f (z) = ; ";:~ ::. Because z^2 sin 7rZ has a zero of order 3
at zo = 0 and 7rcos(7rzo) "# 0, f has a pole of order 3 at zo. By part (iii) of
Theorem 8.2, we have
1 d^2Res[f,O] = - 21 Jim -d 2 7rzcot(7rz)
.z-o z= -
21
lim dd [7r cot (7rz) - 71"^2 z csc^2 ( 7rZ)]
z-+0 Z
= -
21
lim [-7r^2 csc^2 (7rz) - 7r^2 { csc^2 (7rz) - 27rz csc^2 (7rz) cot (?Tz)} J
z-+O
= 71"^2 lim (7rzcot (7rz) -1) csc^2 (7rz)
z-o= 7r^21 1m. 7rzcos(?Tz)--~""""--~""'" - sin(?Tz)
,, .... o sin^3 (7rz)