The Identity Principle For Holomorphic Functions

We prove the Identity Principle for holomorphic functions. Since holomorphic functions have a valid Taylor series near any point where the function is holomorphic, we can use theorems about Taylor series. If a function has a valid Taylor series in a disc, then the function cannot vanish on a set with a limit point unless that function is identically zero. This implies that a holomorphic function on a connected open set cannot vanish on a set with a limit point. This is a very strong and useful theorem.

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