Expressing coefficients of a complex function using integrals

Question

Let $f(z)$ be the function defined by:

\begin{equation} f(z)=a_{-3}z^{-3}+a_{-2}z^{-2}+a_{-1}z^{-1}+a_0+a_1z+a_2z^2+a_3z^3 \end{equation}

How can we express the coefficients $a_i$ using integrals?

This is clearly a rational function $P(z)/Q(z)$ of degree 9. The formula for the residue $a_{-1}$ implies derivatives, not integrals. Maybe a quotient of integrals?

Answer 1

The formula you're looking for is sometimes called the generalized Cauchy integral formula. If $f(z) $ is analytic in a deleted neighbourhood of $z_0$ with Laurent series $\sum_n a_n (z - z_0)^n$,

$$ a_n = \frac{1}{2\pi i} \oint_\Gamma \frac{f(z)\; dz}{(z-z_0)^{n+1}}$$ where $\Gamma$ is a simple positively oriented closed contour around $z_0$ such that $f$ is analytic on and inside $\Gamma$ except possibly at $z_0$. It should be in any complex variables text.

Answer 2

For any $n \in \mathbb{Z}$, $$a_n=\int_0^{1}{f(e^{2i\pi t})e^{-2in\pi t}dt}.$$