evaluate $f(z) $ described in the anticlockwise ( i.e. positive direction)

Question

Evaluate $$\int_{C} \frac{dz}{(z^2+ 4)^2}$$ where $C = \{ z \in \mathbb{C} \mid |z-i| = 2\}$ described in the anticlockwise ( i.e. positive direction)

My attempt: I used Cauchy integral formula

$$\int_{C} \frac{dz}{(z^2+ 4)^2}=\int_{C} \frac{dz}{(z- 2i)^2(z+2i)^2 }= -2\pi i \left.\frac2{(z+ 2i)^3}\right|_{z= 2i}= \frac{\pi}{16}$$

is its correct ?

Answer 1

It is almost correct. The only problem that I was able to detect was that, when you differentiated $\dfrac1{(z+2i)^2}$, you should have obtained $\dfrac{-2}{(z+2i)^3}$, but you seem to have missed that $2$.