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1. The problem statement, all variables and given/known data

Let f(z) be analytic with a zero of order k at z_{0}. Show that f '(z) has a zero of order k-1 at z_{0}.

2. Relevant equations

f(z) has a zero of order k at z_{0}if Ʃc_{n}(z-z_{0})^{n}(here n goes from k to ∞, and k ≠ 0)

3. The attempt at a solution

Well, we of course f '(z) = kc_{k}(z-z_{0})^{k-1}+ (k+1)c_{k+1}(z-z_{0})^{k}+ .... ,

which looks to me like the definition of a function with a zero of order k at z_{0}. ??????

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# Let f(z) be analytic with a zero of order k

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