Let $f$ be a complex function of the form $ f\left( z \right) = \sum\limits_{ - \infty }^\infty {a_j z^j } $
If $f(z) = 0$, $ \forall z:\left| z \right| < 1 $, is $f$ the zero function? (in its domain)
I know that functions that has a power series expansion in every point of it´s domain ( let´s suppose also that the domain it´s a open and connected) vanish on neighborhood of point on a domain if and only if , they are the zero function. It´s true that $ f\left( z \right) = \sum\limits_{ - \infty }^\infty {a_j z^j } $ it´s analytic under this definition?
This is the defition of having a power series expansion in some point.