Hi I'm trying to teach myself Complex analysis and can't find many good examples online. When looking at a past exam paper I stumbled across this question:
Evaluate $$\ \int_C \frac {5z^4-3z^3+2i} {(z-1)^3} dz $$
Where C is any simple closed curve in a counterclockwise direction with z=1 inside C
Any help you can offer would really help out! Thanks
EDIT: $$ f^{(n)}(a) = \frac{2\pi i}{(n-1)!} \oint_C \frac{f(z)}{(z-a)^{n}}\, dz $$ with $$ f(z)=5z^4-3z^3+2i, \quad a=1, \quad n=3. $$ I get the answer to be: $$ \frac{2 \pi i}{2!}f^{(2)}(1)=\pi i (60-18)=42 \pi i $$
Think i may have done this wrong as it seems like an unusually large number?
Hint. One may use Cauchy's integral formula $$ f^{(n)}(a) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z-a)^{n+1}}\, dz $$ with $$ f(z)=5z^4-3z^3+2i, \quad a=1, \quad n=2. $$