I have some past exam questions that I am confused with
http://i39.tinypic.com/vuwxl.png sorry, can't embed images yet
I'm not sure how to approach this, I'm completely lost and just attempted to solve a few:
a) it says $f(z)$ has a pole of order 5, so $ f(z) = \frac{g(z)}{z^5}, g(z)\neq0 $
so then I guess the condition is $a_{4} = \frac{g^{(4)}(0)}{4!}$?
c) $f(\frac{1}{z}) = \frac{g(\frac{1}{z})}{z^5} => f(z) = z^5g(z)$
so the coefficients are $a_{n} = \frac{1}{2\pi i} \oint_\gamma z^5g(z) dz$?
d) $\frac{1}{f(z)} = \frac{g(z)}{z^5} => f(z) = \frac{z^5}{g(z)}$
so, $a_{n} = \frac{1}{2\pi i} \oint_\gamma \frac{z^5}{g(z)} dz$
g) $a_{-1} = \frac{1}{2\pi i} \oint_ \gamma f(z) dz = \frac{1}{2\pi i} = Res(f; c)*I(\gamma; c) = -Res(f; c)$
h) $\frac{a_{n}}{16} = 4^{n}a_{n} => 0 = a_{n}(4^{n} - 4^{-2}) => a_{n} = 0$ or $n = -2$
for e) and f), I'm not sure what the relevance of the essential singularity is
Well, I think you can see I'm clearly lost, would appreciate if you could help me out.