I'm trying to prove the following result: if f is an analytic function defined for all complex numbers, and f(0) = 0 , how can I show that the integral over the unit circle |z| = 1 of z^-n * g(z) is equal to zero for all naturals n , where g(z) = f(1/z)?
I kinda solved it using the Taylor series of f, because it is analytic, and getting the series out of the integral, leaving inside z^-k * z^-n. Then, by Cauchy's Integral Formula, I say that the integral is zero for every combination of k and n, and is equal to 2 * pi * i * 1, being 1 my function in the numerator. I'm almost sure that the integral of z^-n * g(z) must be zero, though. How can I prove it?
Thx in advance