Possible values for a complex integral

May 2010
38
1
Hello,
How many possible values are there for the following integral-

\(\displaystyle \oint_{C}\frac{dz}{(z-z_1)(z-z_2)..(z-z_n)}\)

(I'm guessing the answer is not n, but don't see why)

Should also add that C is a closed contour that does not go through any of the points \(\displaystyle z_i\)
 

shawsend

MHF Hall of Honor
Aug 2008
903
379
Looks like \(\displaystyle 2^n\) to me if the poles are simple and residues do not cancel one or more of each other.
 
  • Like
Reactions: ques and dudyu

Bruno J.

MHF Hall of Honor
Jun 2009
1,266
498
Canada
Hello,
How many possible values are there for the following integral-

\(\displaystyle \oint_{C}\frac{dz}{(z-z_1)(z-z_2)..(z-z_n)}\)

(I'm guessing the answer is not n, but don't see why)

Should also add that C is a simple closed contour that does not go through any of the points \(\displaystyle z_i\)
Otherwise there are infinitely many possible values!
 
May 2010
38
1
Thanks for the replies.
Well, assuming poles are simple and residues don't cancel each other, I take it there are 2 possible values per point \(\displaystyle z_i\) . Why's that?
 

shawsend

MHF Hall of Honor
Aug 2008
903
379
Just start drawing circles around each combination of poles and include one circle that does not include any and keep in mind I would have gotten this one wrong as per Bruno up there. :)