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moo5003
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Homework Statement
Let D be a subset of C and D is open. Suppose a is in D and f:D\{a} -> C is analytic and injective. Prove the following statements:
a) f has in a, a non-essential singularity.
b) If f has a pole in a, then it is a pole of order 1.
c) If f has a removable singularity in a, then theanalytic extension of f to D is one to one too.
Homework Equations
Laurent Series
f(z) = h(1/z) + g(z) on the annulis.
a_n = 0 for all n<k for some k in Z if a is removable
a_n != 0 for infinite n<0 if a is essential
The Attempt at a Solution
a)I'm a little stumped at the moment. I've narrowed my goal down to showing that for some reason because f(z) is one to one that a_n goes to 0. My other guess was to assume that the sing. was essential but I'm not sure how to come up with a contradiction. Any hints you guys can provide would be helpful.
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