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Forums
Mathematics
General Math
MHB Math Problem of the Week
Math POTW for Graduate Students
Analytic Functions with Isolated Zeros of Order k
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[QUOTE="julian, post: 6868593, member: 142346"] [SPOILER] 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)##. [/SPOILER] [/QUOTE]
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Forums
Mathematics
General Math
MHB Math Problem of the Week
Math POTW for Graduate Students
Analytic Functions with Isolated Zeros of Order k
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