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722 CHAPTER 21 Series Representations of Functions

for z in D.Nowlet
∞
1 n+1
F(z) = c n (z − z 0 ) .
n + 1
n=0
It is easy to check that F (z) = f (z) for z in D.

Isolated Zeros

The Taylor expansion of a function about a point gives us important information about the
zeros of the function. A number ζ is a zero of f if f (ζ) = 0. A zero ζ is isolated if there is
an open disk about ζ containing no other zero of f .

For example, sin(z) has isolated zeros at integer multiples of π. By contrast, let

sin(1/z) for z = 0
g(z) =
0 for z = 0.
Then g has zeros at 0 and 1/nπ for each nonzero integer n. 0 is not an isolated zero,
because every disk about 0 contains zeros 1/nπ, which are arbitrarily close to 0 for n suffi-
ciently large. We claim that this behavior of g(z) at 0 prevents g(z) from being differentiable
there.

THEOREM 21.4

Let f be differentiable on a domain G, and let ζ be a zero of f in G. Then either ζ is an isolated
zero or there is an open disk about ζ on which f (z) is identically zero.

This means that a differentiable complex function that is not identically zero on a domain
can have only isolated zeros there.

Proof Write the power series expansion of f about ζ as
∞

f (z) = c n (z − ζ) n
n=0
in some open disk D in G centered at ζ. There are two cases.
First, if every c n = 0, then f (z) = 0 throughout D.
Thus, suppose some coefficients are not zero. Let m be the smallest integer such that c m =0.
Then c 0 = c 1 = ··· = c m−1 = 0, and for z in D,
∞ ∞
n m n
f (z) = c n (z − ζ) = (z − ζ) c n+m (z − ζ) .
n=m n=0
∞ n

Next, let g(z)= c n+m (z −ζ) . Then g is differentiable on D and g(ζ)=c m =0. Furthermore,
n=0
m
f (z) = (z − ζ) g(z).
Because g(ζ) = 0 there is some open disk K about ζ in which g(z) = 0. But then f (z) = 0if z is
in K and is different from ζ. Therefore, ζ is an isolated zero.
If ζ is a zero of f , then the smallest m such that c m = 0 in the Taylor expansion of f about
ζ is called the order of the zero ζ. Because the Taylor coefficients preceding c m must be zero,

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October 14, 2010 15:35 THM/NEIL Page-722 27410_21_ch21_p715-728