Cauchy's Integral Formula/General Result/Corollary

Theorem

Let $\map G z$ be the generating function for the sequence $\sequence {a_n}$.

$\sqbrk {z^n} \map G z := a_n$

Let $\map G z$ be convergent for $z = z_0$ and $0 < r < \cmod {z_0}$.

Then:

$\sqbrk {z^n} \map G z = \ds \frac 1 {2 \pi i} \oint_{\cmod z \mathop = r} \dfrac {\map G z \d z} {z^{n + 1} }$

$\ds \sqbrk {z^n} \map G z$

$=$

$\ds \dfrac 1 {n!} \map {G^{\paren n} } 0$

$\ds $

$=$

$\ds \dfrac 1 {n!} \paren {\dfrac {n!} {2 \pi i} \int_{\partial D} \frac {\map G z} {z^{n + 1} } \d z}$

$\ds $

$=$

$\ds \frac 1 {2 \pi i} \oint_{\cmod z \mathop = r} \dfrac {\map G z \d z} {z^{n + 1} }$

$\blacksquare$