280 CHAPTER 7 • TAY1-0R AND LAURENT SERIES
An immediate oonsequence of Theorem 7. 11 is Corollary 7.4. The proof is
left as a.n exercise.t Corollary 7 .4 If f (z) and g (z) are analytic at z = a and have zeros of orders
m and n, respectively, at z =a, then their product h (z) = f (z) g (z) has a zero
of order m + n at z = a. •
•EXAMPLE 7.11 Let f (z) = z^3 sin z. Then f(z) can be factored as t he
product of z3 and sin z, which have zeros of orders m = 3 and n = 1, respectively,
a t z = 0. Hence z = 0 is a zero of order 4 off (z).
Theorem 7.12 gives a useful way to characterize a pole.