Suppose that $f$ is analytic in a simply connected domain $D$ containing distinct points $z_1, z_2 ,\ldots,z_n $ and that $\gamma$ is simple closed curve enclosing $z_1, z_2 ,\ldots,z_n $.
Set $w(z)= \prod_{k=1}^n (z-z_k)$ . Prove that
$P(z) =\int_{\gamma} \frac {f(\zeta)}{w(\zeta)} \frac{w(\zeta)-w(z)}{\zeta-z}d\zeta$ defines a polynomial of degree $n-1$ satisfying $P(z_k) =f(z_k), k =1,2,3,\ldots,n.$
I don't know what thought I am suppose to present here. What I just know is to do the partial fraction of the $w(z)$. Then I just stuck not knowing the fact I am using here. This is actually a recent comprehensive exam question which I could not solve. My professor gave me some hint but that also did not help. The idea of partial fraction came out of his mouth. But I don't remember what he told me to do to complete the problem. I really wish to see the detail solution of the problem. Thanks in advance.
Addendum
Partial Fraction of $\frac{1}{w(\zeta)}= \frac {1}{\prod _{k=1}^n (\zeta-z_k)}$ is given by $\frac {A_1}{\zeta-z_1}+\frac {A_2}{\zeta-z_2}+\cdots+\frac {A_n}{\zeta-z_n}$
Where each $A_i = \frac{1} {\prod_{j=1}^n z_i-z_j} , j \neq i$
Sorry I missed that not equal to part, I don't know how to accomodate that in the product, any edit appreciated.