Let be a series $ \sum\limits_{ - \infty }^\infty {a_jz^j} $. Convergent on $1<|z|<4$. Such that vanishes on $2<|z|<3$ It's true that all the coefficients $a_j$ are zero?
I know by the principle of analytic continuation, that this would be true, if I know that the series has the form $ \sum\limits_{ 0 }^\infty {b_jz^j} $. Here I don't know how to do. Maybe in the convergence zone, I can write the series on that form, but I don't know how :/