# Complex analysis proof with residue theorem, argument principle

1. Apr 30, 2013

### nate9228

1. The problem statement, all variables and given/known data

Let C be a regular curve enclosing the distinct points w1,..., wn and let p(w)= (w-w1)(w-w2)...(w-wn). Suppose that f(w) is analytic in a region that includes C. Show that P(z)= (1/2$\pi$i)∫(f(w)$\div$p(w))$\times$((p(w)-p(z)$\div$(w-z))$\times$dw
is a polynomial of degree n-1 with P(wk) = f(wk), k= 1,2,...

2. Relevant equations

3. The attempt at a solution
So far I know this has something to do with the argument principle and possibly the residue theorem. I believe the inside of the integral reduces to (p'(w)$\div$p(w))$\times$f(w)dw, which is why I think the argument principle pertains to this problem. After this I am not sure what to do. This problem is from Bak and Newman Complex Analysis, third edition, chapter 10, if anyone is familiar with the book.

2. Apr 30, 2013

### nate9228

Wow, the code did not turn out at all how I thought it was going to. I apologize for this confusion; I used the symbols button and just assumed they would translate to standard notation.