Past exam of the mathematics course of the University of Cambridge / 2001 / ib / Paper 1 / 7E / Solution

The Cauchy integral formula states that if $f$ is holomorphic on a neighbourhood of a simple closed positively oriented contour and its interior, then, for an interior point $z$,Fix $0<r<1$ and apply it on $|\xi |=r$. For $|z|<r$, the geometric seriesconverges uniformly in $\xi$ and locally uniformly in $z$. Indeed, for $|z|\le s<r$ its absolute terms are bounded by $s^n/r^{n+1}$. Since $f$ is bounded on the contour, termwise contour integration is valid, givingTermwise differentiation of this locally convergent power series gives $f^{(n)}(0)=n!c_n$. ThereforeThis also derives the needed Taylor series representation rather than assuming an unjustified interchange at the boundary of the analytic disk.