Cauchy coefficient formula ($[z^n]f(z)=(2\pi i{)}^{-1}\oint f(z)z^{-n-1}dz$)

For a function holomorphic on a disk containing the contour, its Taylor series converges uniformly on the integration circle. Integrating termwise leaves only the term with exponent $-1$, giving the displayed coefficient extraction. The power is $-n-1$, not $-n$; the latter extracts the coefficient at level $n-1$.