# Irregular singular point

A concept that arose in the analytic theory of linear ordinary differential equations. Let $A(t)$ be an $(n\times n)$-matrix that is holomorphic in a punctured neighbourhood of $t_0\neq\infty$ and that has a singularity at $t_0$.
The point $t_0$ is then called a singular point of the system
$$\dot x=A(t)x.\label{*}\tag{*}$$
There are two non-equivalent definitions of an irregular singular point. According to the first one, $t_0$ is called an irregular singular point of \eqref{*} if $A(t)$ has a pole of order greater than one at $t_0$ (cf. Analytic theory of differential equations, as well as ). According to the second definition, $t_0$ is called an irregular singular point of \eqref{*} if there is no number $\sigma>0$ such that every solution $x(t)$ grows not faster than $|t-t_0|^{-\sigma}$ as $t\to t_0$ along rays (cf. ). The case $t_0=\infty$ can be reduced to the case $t_0=0$ by the transformation $t\to t^{-1}$. An irregular singular point is sometimes called a strongly-singular point (cf., e.g., Bessel equation). In a neighbourhood of an irregular singular point the solutions admit asymptotic expansions; these were studied by H. Poincaré for the first time .