View source for Irregular singular point

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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|Analytic theory of differential equations]], as well as [[#References|[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. [[#References|[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|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 [[#References|[1]]].

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