Monodromy eigenfunction Laurent representation

For a linear ordinary differential equation with holomorphic coefficients on a punctured disc, analytic continuation of a basis gives an invertible monodromy matrix. A nonzero eigenvector with eigenvalue $\lambda =e^{2\pi i\sigma }$ selects a solution $w$ for which $z^{-\sigma }w$ is single-valued and holomorphic. Its Laurent series gives $w=z^{\sigma }{\sum }_{n\in \mathbb{Z}}{c}_n{z}^n$. No regular-singular hypothesis is needed; infinitely many negative powers may occur.