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.