Cauchy integral formula or something else?

Question

I need to determine the function $\;f(z)$ if $$f''(z)=\oint_{\partial C_1(0)}{\sin^2\xi \over\left(\xi-z\right)^3}\mathbb{d}\xi$$

with $C_1(0):\left|z\right|<1$ positive. Additionally $\;f(0)=0,\;f'(0)=2$.

I am stuck on the integral. Quite obvious they want me to apply Cauchy's integral formula. However I am not sure how I can handle this with the integrand $\xi$. Is there some substitution to be made or anything else?

Answer 1

Hint: Just rename variables, does $$f''(a)=\oint_{\partial C_1(0)}{\sin^2z \over\left(z-a\right)^3}\mathbb{d}z$$ look at all familiar?

Answer 2

compare $$f''(a)=\oint_{\partial C_1(0)}{\sin^2z \over\left(z-a\right)^3}\mathbb{d}z$$ and $$f''(a)=\frac{2!}{2 \pi i}\oint_{\partial C_1(0)}{f(z) \over\left(z-a\right)^3}\mathbb{d}z$$