The Grassmann analytic continuation principle says that a real function $f(x,\xi)$ that depends on $n$ real variables $x=(x_1,\ldots,x_n)$ and $n$ nilpotent Grassmann numbers $\xi=(\xi_1,\ldots,\xi_n)$ has the following Taylor expansion containing only a finit ($2n?$) number of terms

$$ f(x+\xi) = \sum_{\alpha}\frac{\xi^{\alpha}}{\alpha!}\frac{\partial^{\alpha}}{\partial x^{\alpha}}f(x) $$

where the $\xi_j \in \mathcal{G}_n$ which is the fermionic part of the supermanifold at hand are given by

$$ \xi_j =\sum_{|\epsilon|>0}a_{\epsilon}^j\theta^{\epsilon}$$

The $\theta_i$ are the generators of $\mathcal{G}_n$, $\alpha=(\alpha_1,\ldots,\alpha_n)\in\mathbb{N}^n$, and

$\xi^{\alpha}=\xi_1^{\alpha_1}\ldots\xi_n^{\alpha_n}$.

How is this related to expanding a superfield in Grassmann coordinates?

For example, a a scalar superfiled $\mathcal{F}(x,\theta_1,\theta_2)$ that depends on the ordinary

spacetime coordinates as well as on two Grassmann coordinates can be expanded as

$$\mathcal{F}(x,\theta_1,\theta_2) = A(x)+B(x)\theta_1+C(x)\theta_2+D(x)\theta_1\theta_2$$

Comparing this expansion to the Grassmann analytic continuation principle above I would think that in this specific case we have $n=2$, $\alpha=(\alpha_1,\alpha_2)$ and $\xi=\xi_1^{\alpha_1}\xi_2^{\alpha_2}$ as there are two Grassmann coordinates in addition to conventional spacetiem.

The first thing I dont understand is why in the Grassmann analytic continuation principle the function is not expanded directly in the $\theta_i$ but the $\xi$ are used instead which makes it hard for me to see what is going on.

Otherwise, I would have guessed that from setting $\alpha_1=0$ and $\alpha_2=0$ the coefficient $A(x)$ would correspond to $f(x)$, from setting $\alpha = (2,1)$ one obtains $B(x)=\frac{\partial f}{\partial\theta_1}$, from setting $\alpha = (1,2)$ $C(x)=\frac{\partial f}{\partial\theta_2}$ and from $\alpha = (1,1)$ one has $D(x)=\frac{\partial^2 f}{\partial\theta_1\partial\theta_2}$