Let \(P(z)=\sum_{n=0}^{\infty} a_n(z-z_0)^n\) be a convergent power series , i.e. with positive radius of convergence. Due to the differentiability of the power series we obtain

\begin{equation*} P^{(k)}(z)=\sum_{n=k}^{\infty} n(n-1)\cdots (n-k+1)a_n(z-z_0)^{n-k}. \end{equation*}

Thus, the coefficients are given by

\begin{equation*} a_n=\frac{P^{(n)}(z_0)}{n!}. \end{equation*}

For the multidimensional case we have

\begin{equation*} a_\mu = \frac{\partial^\mu f(z_0)}{\mu!}. \end{equation*}

[1, Proposition 2.2.3]

References Link to heading

1. S. Krantz and H. Parks, A Primer of Real Analytic Functions. Boston, MA: Birkhäuser Boston, 2002. doi:10.1007/978-0-8176-8134-0