(a) Let $f$ be a holomorphic function in $|z| < 2$ such that (*) $\int_{|z| = 1} \frac{f(z)}{nz-1} = 0$ for each $n = 2, 3, ....$ Prove that $f(z) \equiv 0$ in $B(0; 2)$.
(b) If we only assume that $f$ is a holomorphic function defined in $0 < |z| < 2$ such that (*) holds, can we still draw the conclusion that $f \equiv 0$ ?
I have managed to solve part (a) by using Cauchy's Integral Formula as well as the Identity Theorem.
However, how to go about solving part (b)? I have tried to use the Laurent series expansion but I wouldn't know whether I am in the right direction.