7.5 • APPLICATIONS OF TAYLOR AND LAURENT SERIES 2857.5 Applications of Taylor and Laurent Series.
In this section we show how you can use Taylor and Laurent series to derive
important properties of analytic functions. We begin by showing that the zeros
of an analytic function must be isolated unless the function is identically zero.
A point a of a set Tis called isolated if there exists a disk DR (a) about a that
does not contain any other points of T.
The proofs of the following corollaries are given as exercises.t Corollary 7.9 Suppose t hat f is analytic in the domain D and t hat a E D. If
there exists a sequence of points {zn} in D such that Zn --+ a, and f (zn) = 0,
then f (z) = 0 for all z E D. •
t Corollary 7. 10 Suppose that f and g are analytic in the domain D , where
a E D. If there exists a sequence { zn} in D s uch that Zn -+ a, and f (zn) = g (Zn)
for all n , then f (z) = g (z) for all z ED. •