Possible values for a complex integral

[dudyu]

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]

Looks like \(\displaystyle 2^n\) to me if the poles are simple and residues do not cancel one or more of each other.

 

[Bruno J.]

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!

 

[dudyu]

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]

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.

 

[dudyu]

Got it. Thanks very much!