### Analytic Zeros Proof

Let f be analytic on a domain D. Prove that if f(z(0))=0 and if f is not identically zero, then z(0) is a zero of f of some finite order m.

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Let f be analytic on a domain D. Prove that if f(z(0))=0 and if f is not identically zero, then z(0) is a zero of f of some finite order m.

For the following expression, give one interpretation that makes it true and one interpretation that makes it false: {see attachment for expression}

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Respond/pick up the credit if you absolutely know the solution is correct. If you can make an improvement on the solution in correctness, clarity, presentation, or if a proof can be more elegant, than please rewrite the entire solution.

If the solution to this nonnegative integer question is correct, then you may respond that it is. If the solution needs ANY kind of improvement, in presentation, in clarity, in correctness, if a proof can be more elegant, then please rewrite the entire solution.

Suppose that f: C->C and that f is analytic at a point z0 element of C. Prove that there exists a real number r>0 such that, the nth derivative of z0=[n!/(2 pi r^n)]x[int(e^(-niy)f(z0+re^(iy)) from 0 to 2pi with respect to y for all n element of Natural numbers.

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