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 [2]). 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. [3]). 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 [1].

References

[1]	H. Poincaré, "Sur les intégrales irrégulières des équations linéaires" Acta Math. , 8 (1886) pp. 295–344
[2]	W. Wasov, "Asymptotic expansions for ordinary differential equations" , Interscience (1965)
[3]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17