We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity, and $\omega(t) = e^{2i\pi t}$ for $t \in [0,1]$. Let $f \in \mathcal{E}$ whose power series expansion we denote $\sum a_{n} z^{n}$. Show that, for all $k \in \mathbb{Z}$:
$$\int_{0}^{1} f(\omega(t)) \omega(t)^{-k} \,\mathrm{d}t = \begin{cases} a_{k} & \text{if } k \in \mathbb{N} \\ 0 & \text{otherwise} \end{cases}$$