The answer to this question shows that if I have a real nonsingular matrix $M$, such that its Taylor expansion in $\epsilon$ is
$$M(x+\epsilon)= \sum_{n=0}^\infty M_n(x) \epsilon^n $$
its inverse can be written as an expansion using the following formula
$$B = \sum_{i=0}^\infty b_n\epsilon^n$$
$$b_0 = a_0^{-1},$$ $$b_1 = -a_0^{-1}a_1a_0^{-1}$$ $$b_2 = a_0^{-1}a_1a_0^{-1}a_1a_0^{-1} -a_0^{-1}a_2a_0^{-1}$$ $$b_3 = - a_0^{-1}a_1a_0^{-1}a_1a_0^{-1}a_1a_0^{-1} + a_0^{-1}a_1a_0^{-1}a_2a_0^{-1} + a_0^{-1}a_2a_0^{-1}a_1a_0^{-1} - a_0^{-1}a_3a_0^{-1}$$
My question is: how does this change if I also have negative powers of $\epsilon$ in the expansion? i.e.
$$\mathcal{M}(x)=\sum_{n=-\infty}^\infty \mathcal{M}_n(x)\epsilon^n$$
If you define $M(x)$ as a Cauchy principal value, that is, $M(x)=\lim_{k\rightarrow +\infty}\sum_{n=-k}^k M_n(x)\epsilon^n$, then
$M(x)=\sum_{n=0}^{+\infty}U_n(x)\epsilon^n$ where $U_0=M_0,U_{2p}=M_{2p}+M_{-2p},U_{2p+1}=M_{2p+1}-M_{-2p-1}$.