Cauchy's Integral Formula/General Result/Corollary

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Theorem

Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$.

Let the coefficient of $z^n$ extracted from $G \left({z}\right)$ be denoted:

$\left[{z^n}\right] G \left({z}\right) := a_n$

Let $G \left({z}\right)$ be convergent for $z = z_0$ and $0 < r < \left\lvert{z_0}\right\rvert$.


Then: $\left[{z^n}\right] G \left({z}\right) = \displaystyle \frac 1 {2 \pi i} \oint_{\left\lvert{z}\right\rvert \mathop = r} \dfrac {G \left({z}\right) \rd z} {z^{n + 1} }$


Proof

We have that:

$\displaystyle \oint_{\left\lvert{z}\right\rvert \mathop = r} z^m \rd z = 0$ for all integers $m$ excapt for $m= -1$.

In that case:

$\displaystyle \int_{-\pi}^\pi \left({r e^{i \theta} }\right)^{-1} \rd \left({r e^{i \theta} }\right) = i \int_{-\pi}^\pi \rd \theta = 2 \pi i$



Sources