How can I use Cauchy formula to this Integral?

Question

$$∫ \frac{5z^2 - 3z + 2}{(z-1)^3} dz$$

and the contour is any closed simple curve involving z=1 (sorry, I forgot to write this information)

Need to solve it using Cauchy Integral formula

Can anyone explain this to me? Thanks!

Answer 1

You can consider your cantour as a circle centered at one with radius $r$. Then the integral equals

$$\int_{C} \frac{f(z)}{(z-1)^3}dz = \frac{2\pi i}{2!} f^{(2)}(1)$$

Note:

Theorem 1: If $f(z)$ is analytic inside and on the boundary $C$ of a simply-connected region $R$ and $a$ is any point inside $C$ then

$$ f(a)=\frac{n!}{2\pi i}\int_{C} \frac{f(z)}{(z-a)^{n+1}}dz. $$