The Identity theorem – do the accumulation point need to be inside of S or inside of D

complex-analysis

Area: The Identity Theorem (Complex Analysis)

Let D be a non-empty, connected open subset of the Complex Plane and let f be analytic on D. Then the following three statements are equivalent.

  1. f(z) = 0 for all z in D.
  2. The set $ S = \{z\in D\mid f(z)=0\}$ contains an accumulation point.
  3. There exists a point $z_0 \in D$ such that $f^{(n)}(z_0) = 0 \ $ for all $n \geq 0$.

Question: I have been wondering about the second statement. Some sources state that the accumulation point of the set S needs to be contained inside of S meanwhile others just state that it needs to be contained inside of D. Which one is right?

I think that this theorem comes from the following: If f is analytic on a domain D and is not equal to zero on the whole domain then the zeros of f is discrete in D. Hence the set of all zeroes for an analytic function are isolated and the set of them can therefore not have a limit point (accumulation point). The only way the set of all zeros (S in the identity theorem) can have a limit point would be if the function is equal to 0 on the whole domain.

Best Answer

Some sources state that the accumulation point of the set S needs to be contained inside of S meanwhile others just state that it needs to be contained inside of D.

It makes no difference. If $S = \{z\in D\mid f(z)=0\}$ has an accumulation point $z_0 \in D$ then $f(z_0) = 0$ because $f$ is continuous, so that $z_0 \in S$.

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