# Irregular singular point

A concept that arose in the analytic theory of linear ordinary differential equations. Let be an -matrix that is holomorphic in a punctured neighbourhood of and that has a singularity at .

The point is then called a singular point of the system

(*) |

There are two non-equivalent definitions of an irregular singular point. According to the first one, is called an irregular singular point of (*) if has a pole of order greater than one at (cf. Analytic theory of differential equations, as well as [2]). According to the second definition, is called an irregular singular point of (*) if there is no number such that every solution grows not faster than as along rays (cf. [3]). The case can be reduced to the case by the transformation . 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 |

**How to Cite This Entry:**

Irregular singular point.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Irregular_singular_point&oldid=18547