Complex analysis proof with residue theorem, argument principle

  • Thread starter nate9228
  • Start date
  • #1
42
0

Homework Statement



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,...

Homework Equations





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.
 

Answers and Replies

  • #2
42
0
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.
 

Related Threads on Complex analysis proof with residue theorem, argument principle

Replies
2
Views
770
  • Last Post
Replies
1
Views
3K
Replies
5
Views
4K
Replies
1
Views
821
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
3
Views
2K
Replies
3
Views
1K
  • Last Post
Replies
2
Views
3K
Top