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Elle
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 Analytic functions   « on: Dec 2nd, 2005, 6:11am » Quote Modify Remove

I found the following problem in a complex analysis book and that's why I'm writing on this forum even if it's not really a complex analysis problem

Suppose f: R -> R is continuous, f^2 is real analytic and f^3 is real analytic. Prove that f is real analytic.
Warning: beware of the zeros of f.

All help is greatly appreciated as I'm getting a bit frustrated..
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Icarus
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 Re: Analytic functions   « Reply #1 on: Dec 2nd, 2005, 3:54pm » Quote Modify

I don't have time to say more, but
(1) all real analytic functions are complex analytic functions restricted to the real line.
(2) f = f3/f2. So all you have to do is show that the zeros of f2 induce only removable singularities in f. This is because they also must be zeros of f3. All you need to do is show that they are zeros of lesser order.
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MonicaMath
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 Re: Analytic functions   « Reply #2 on: Mar 2nd, 2009, 10:46am » Quote Modify

so how we can do this .... ??!! can u help us more ,,,,

thank you
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Eigenray
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 Re: Analytic functions   « Reply #3 on: Mar 2nd, 2009, 12:52pm » Quote Modify

We have two analytic functions g and h such that g3 = h2.  What can you say about their orders at a zero?
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