  In this paper, the numerical differentiation by integration method based on
Jacobi polynomials originally introduced by Mboup, Fliess and Join is revisited
in the central case where the used integration window is centered. Such method
based on Jacobi polynomials was introduced through an algebraic approach and
extends the numerical differentiation by integration method introduced by
Lanczos. The here proposed method is used to estimate the $n^{th}$ ($n \in
\mathbb{N}$) order derivative from noisy data of a smooth function belonging to
at least $C^{n+1+q}$ $(q \in \mathbb{N})$. In the recent paper of Mboup, Fliess
and Join, where the causal and anti-causal case were investigated, the
mismodelling due to the truncation of the Taylor expansion was investigated and
improved allowing a small time-delay in the derivative estimation. Here, for
the central case, we show that the bias error is $O(h^{q+2})$ where $h$ is the
integration window length for $f\in C^{n+q+2}$ in the noise free case and the
corresponding convergence rate is $O(\delta^{\frac{q+1}{n+1+q}})$ where
$\delta$ is the noise level for a well chosen integration window length.
Numerical examples show that this proposed method is stable and effective.
