Argument Theorem - Complex Analysis

[Shay10825]

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

Evaluate (1/2ipi)* contour integral of [z^(n-1)] / [(3z^n) - 1 ] dz

2. Relevant equations

I would assume you would have to use the Argument Theorem since this problem comes after the proof of the argument theorem in my book.


3. The attempt at a solution

z^(n-1) has (n-1) zeros
(3z^n) - 1 has n zeros

therefore the integral is equal to (n-1)-n = -1

Is this correct?

Thanks

[Billy Bob]

Yes, use the Argument Principle, but no, you did it incorrectly.

If you are integrating f'(z)/f(z), then count the zeros (n) and poles (none) of f(z) only, not f'(z). Also, if f(z)=3z^n-1, then f'(z)=3nz^(n-1) so don't forget to take the 3n into account.

I guess you are assuming the contour is a simple closed contour enclosing all the roots of f(z).