I have some past exam questions that I am confused with






I'm not sure how to approach this, I'm completely lost and just attempted to solve a few:

a) it says $\displaystyle f(z)$ has a pole of order 5, so $\displaystyle f(z) = \frac{g(z)}{z^5}, g(z)\neq0 $

so then I guess the condition is $\displaystyle a_{4} = \frac{g^{(4)}(0)}{4!} $? But that's just applying the formula for the coefficients...

c) $\displaystyle f(\frac{1}{z}) = \frac{g(\frac{1}{z})}{z^5} => f(z) = z^5g(z) $

so the coefficients are $\displaystyle a_{n} = \frac{1}{2\pi i} \oint_\gamma z^5g(z) dz $?

d) $\displaystyle \frac{1}{f(z)} = \frac{g(z)}{z^5} => f(z) = \frac{z^5}{g(z)} $

so, $\displaystyle a_{n} = \frac{1}{2\pi i} \oint_\gamma \frac{z^5}{g(z)} dz $

g) $\displaystyle a_{-1} = \frac{1}{2\pi i} \oint_ \gamma f(z) dz = \frac{1}{2\pi i} = Res(f; c)*I(\gamma; c) = -Res(f; c)$

h) $\displaystyle \frac{a_{n}}{16} = 4^{n}a_{n} => 0 = a_{n}(4^{n} - 4^{-2}) => a_{n} = 0 $ or $\displaystyle 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.