I am currently trying to evaluate the derivatives of a function $F(z)$ in z=0, which is only known numerically on the unit circle ("$UC$") in the complex plane. My question is this:
given $z=\mathrm{e}^{-iwT}$ on $UC$ and Cauchy's integral formula, we get
\begin{equation} F^{(n)}(z=0) = \frac{1}{2\pi i}\int_{UC} \frac{F(z)}{z^{n+1}}\mathrm{d}z = -\frac{\omega}{2\pi}\int_0^{2\pi/\omega}F(\mathrm{e}^{-i\omega T})\mathrm{e}^{in\omega T} \mathrm{d}T\\ =-\frac{1}{2\pi}\int_0^{2\pi}F(\mathrm{e}^{-i\tilde{T}})\mathrm{e}^{in\tilde{T}} \mathrm{d}\tilde{T}. \end{equation} Does anybody know of any quadrature schemes fitting for evaluating the above expression without too many sampling points? Im thinking there might be something equivalent to e.g. Gauss-Laguerre quadrature, but with another weight-function? Any suggestions would be appreciated.
For anybody curious about what it is for, the derivatives of $F$ in $z=0$ are actually Greens functions of a perturbed quantum system in the adiabatic limit. The order of derivative is related to how many photons are absorped/emitted. $F$ is actually matrix valued and will consume a lot of RAM, which is why i would like to keep the sampling points to a minimum. $F$ is formally a Laurent-series on the form $F(z,E) = \sum _{n=-\infty}^{\infty}z^n G_n(E)$, where the $G_n$'s are various Greens functions of the system.
The trapezoidal rule would be ideal here, because the function is periodic (as well as smooth) and this rule converges almost exponentially fast for such functions.