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Complex analysis proof with residue theorem, argument principle

  1. Apr 30, 2013 #1
    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[itex]\pi[/itex]i)∫(f(w)[itex]\div[/itex]p(w))[itex]\times[/itex]((p(w)-p(z)[itex]\div[/itex](w-z))[itex]\times[/itex]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)[itex]\div[/itex]p(w))[itex]\times[/itex]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. jcsd
  3. Apr 30, 2013 #2
    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.
     
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