Could someone please assist me with the second part of the second paragraph, from "By expanding $f_1$..."?
I am not convinced that my expansion for $f_1$ is right - I used the standard binomial, but with $$(z^2 + 4 \pi^2 r^2)^{-1} = z^{-2}(1 + ({2\pi r \over z})^2)^{-1}$$ then normal binomial on this, but only need the first term. I could then find the Taylor series without using my previous Laurent series for $f$, which I assume that I'm not supposed to do since I am then finding the Laurent series for $f$, and my previous was only valid for $ 0 < |z| < 2\pi $.
Thanks in advance! :)