Let $N$ be a positive integer and $a_n$ be a complex number for every $-N \leq n \leq N$. Consider the holomorphic function on $\{z \in \mathbb{C} \mid z \neq 0\}$ given by $$F(z) = \sum_{n=-N}^{n=N} a_n z^n$$ Consider the function $f$ defined on the open unit disc $\{z \in \mathbb{C} : |z| < 1\}$ by $$f(z) = \frac{1}{2\pi i} \int_{\Gamma} \frac{F(\xi)}{\xi - z}\,d\xi$$ where $\Gamma$ is the boundary of the disc, oriented counterclockwise. Write down an expression for $f$ in terms of the coefficients $a_n$ of $F$.
Let $N$ be a positive integer and $a_n$ be a complex number for every $-N \leq n \leq N$. Consider the holomorphic function on $\{z \in \mathbb{C} \mid z \neq 0\}$ given by
$$F(z) = \sum_{n=-N}^{n=N} a_n z^n$$
Consider the function $f$ defined on the open unit disc $\{z \in \mathbb{C} : |z| < 1\}$ by
$$f(z) = \frac{1}{2\pi i} \int_{\Gamma} \frac{F(\xi)}{\xi - z}\,d\xi$$
where $\Gamma$ is the boundary of the disc, oriented counterclockwise. Write down an expression for $f$ in terms of the coefficients $a_n$ of $F$.