3
$\begingroup$

Given $$\int_{\gamma}\frac{1}{(z-a)(z-\frac{1}{a})}dz,$$ and $0, where $\gamma(t)=e^{it}$ and $0\le t \le 2\pi$

I am trying to find the residue of$$f(z)=\frac{1}{(z-a)(z-\frac{1}{a})}$$

The answer says $\operatorname{res}(f,\mathbb C)=\frac{1}{(a-\frac{1}{a})}$?

Any help will be appreciated, thanks.

  • 0
    Residues are defined at a pole of finite order. So, this is a function from the poles of a meromorphic function to the complex numbers. Are you attempting to compute an integral over a path from the residue2012-01-19
  • 0
    Is your ultimate goal to compute that integral? If so, why are you computing the residues at both singularities? Do both singularities contribute to the value of the integral?2012-01-19
  • 0
    @Santiago, Hi, yes my ultimate goal is to compute the integral. Sorry I just started learning residues so Im not good at these. So are you saying I can just take $\frac{1}{(a-\frac{1}{a})}$? How would I know which singularities contribute to the value of the integral?2012-01-19
  • 0
    @Thomas, no need to apologize. Which version of the "Residue Theorem" are you trying to use? The answer to your problem is provided by ncmathsadist below, but if you're just starting out, you may not know what "winding number" means.2012-01-19
  • 1
    Just one more thing to think about: why is the restriction that 0 < a < 1 important?2012-01-19

4 Answers 4