 Solution

The complex Taylor series theorem states that if $f$ is holomorphic in $|z-z_0|<R$, thenThe power series converges absolutely there and uniformly on each smaller closed disc; its coefficients are unique. For $|z-z_0|<r<R$, the Cauchy integral formula also gives $f^{(n)}(z_0)/n!=(2\pi i{)}^{-1}{\int }_{|\zeta -z_0|=r}f(\zeta )(\zeta -z_0{)}^{-n-1}d\zeta$. This is the analytic, rather than finite real-variable, form of Taylor theorem relevant here.