Suppose that $f$ is analytic on a disk $B(a;r)$ except for finitely many points $a_1 , a_2 , \cdots , a_n$ such that $\lim_{z \rightarrow a_i } (z - a_i ) f(z) = 0$ for $i = 1,2, \cdots, n$. Then show that $\int_{\gamma} f(\zeta)\ d\zeta = 0$ for any simple closed contour $\gamma$ on $B(a;r) \setminus \{a_1 , a_2 , \cdots , a_n \}$.
I have proved this result for $n=1$ which is easy to prove using Cauchy's deformation theorem and estimation theorem. But I have failed to generalize it for any finite case. Please help me.
Thank you in advance.