- #1
Euge
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Suppose ##f## is analytic in an open set ##\Omega \subset \mathbb{C}##. Let ##z_0\in \mathbb{C}## and ##r > 0## such that the closed disk ##\mathbb{D}_r(z_0) \subset \Omega##. If ##f## has a zero of order ##k## at ##z = z_0## and no other zeros inside ##\mathbb{D}_r(z_0)##, show that there an open disk ##D## centered at the origin such that for all ##\alpha\in D##, ##f## takes on the value ##\alpha## exactly ##k## times, counting multiplicity.