Power Series, Isolated Zeros

Isolated Zeros

Every zero is surrounded by a neighborhood in which f is nonzero. Let f(0) = 0 for convenience. If this zero has infinite order, i.e. if all its derivatives are 0, then f is identically 0 everywhere. That's not very interesting, so set this case aside.

Let f have an m^th order zero at 0 and write f = z^mg, where g(0) is nonzero. By continuity g is nonzero on a neighborhood containing 0, and z^m is nonzero everywhere except 0, hence f is nonzero on a neighborhood about 0. The zeros of an analytic function are isolated from each other.

If f is analytic on and inside a simple closed curve, or any other closed bounded set in the plane, it can have only a finite number of zeros. Suppose the opposite, and let c be a zero that is a cluster point for the set of zeros. This contradicts the fact that zeros are isolated.

An analytic function on an open domain could have infinitely many zeros, as shown by cos(1/(1-z)) on the open unit disk.