Can you help me solve the following problem?
Let $f$ be a holomorphic function on $D_{r}(0), $ the disc of radius $r>1$ with center on the origin. Calculate the following integral:
$\int_{|z|=1} (2\pm(z + z^{-1}))\frac{f(z)}{z}\mathbb dz$
Solution:
Observe that $(2\pm(z + z^{-1}))\frac{f(z)}{z}= \frac{\left( 2z \pm z^{2} \pm 1 \right)f(z)}{z^{2}}$. Let $h(z)= \left( 2z \pm z^{2} \pm 1 \right)f(z)$. Then this function is holomorphic in $D_{r}(0)$ since $f$ is and it is multiplied by a polynomial. Then by the Cauchy's Integral Formula:
\int_{|z|=1} (2\pm(z + z^{-1}))\frac{f(z)}{z}\mathbb dz =\int_{|z|=1} \frac{h(z)}{z^{2}} \mathbb dz = 2\pi i \frac{h'(0)}{1!}
But h'(0) = 2f(0) \pm f'(0).