Analytic Functions with Isolated Zeros of Order k

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In summary, an analytic function with isolated zeros of order k is a complex-valued function that is differentiable at every point in its domain, except at a finite number of isolated points where it has a zero of order k. This is different from regular zeros, which occur continuously throughout the domain. The order of a zero directly affects the behavior of an analytic function, with higher order zeros resulting in a more "rounded" function near that point. An analytic function can only have a finite number of isolated zeros, and they are used in various areas of mathematics and science for understanding functions and solving equations and modeling physical systems.
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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.
 
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Is the final disk really centered at 0 and not ##z_0##?

Edit; never mind, I'm bad at reading.
 
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We wish to show that

\begin{align*}
f (z) - \alpha = 0
\end{align*}

happens exactly ##k## times, counting multiplicity inside ##\mathbb{D}_r (z_0)## for all ##\alpha## such that ##|\alpha| < R## for some ##R >0##.

Rouche's theorem:

"Let ##f## and ##g## be analytic in a simply connected domain ##U \in \mathbb{C}##. Let ##C## be a simple closed contour in ##U##. If ##|f(z)| > |g(z)|## for every ##z## on ##C##, then the functions ##f(z)## and ##f(z) + g(z)## have the same number of zeros, counting multiplicities, inside ##C##."

Take ##C## to be the circle centred at ##z_0## with radius ##r##. Note ##|f(z)| \not= 0## on ##C##. Let ##R = \min_C |f(z)|## and define an open disk ##D## about the origin of radius ##R##. For ##\alpha \in D##, write ##g(z) = - \alpha##. Then ##|f(z)| > |g(z)|## for every ##z## on ##C##. By Rouche's theorem ##f(z)## and ##f(z) + g(z)## have the same number of zeros, counting multiplicities, inside ##C##. Therefore, ##f## takes on the value ##\alpha## exactly ##k## times, counting multiplicity, inside ##\mathbb{D}_r (z_0)##.
 
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