# Regular singular point

A notion in the theory of ordinary linear differential equations with an independent complex variable. A point is called a regular singular point of the equation

(1) |

or of the system

(2) |

with analytic coefficients, if is an isolated singularity of the coefficients and if every solution of (1) or (2) increases no faster than for some , as tends to within an arbitrary acute angle with vertex . This last restriction is necessary in view of the fact that in a neighbourhood of a regular singular point the solutions are non-single-valued analytic functions, and as along an arbitrary curve, they can increase essentially faster than they do when over a ray with vertex .

For a singular point of the coefficients of (1) or (2) to be a regular singular point of (1) or (2), it must be a pole (of a function), and not an essential singular point, of the coefficients. For equation (1) there is Fuchs' condition: The singular point of the coefficients of equation (1) is a regular singular point of (1) if and only if the functions , , are holomorphic at zero. In the case of the system (2) there is the following sufficient condition: If the entries of the matrix have a simple pole at a point , then this point is a regular singular point of (2).

#### References

[1] | V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) |

[2] | E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17 |

[3] | A.H.M. Levelt, "Hypergeometric functions I-IV" Proc. Koninkl. Nederl. Akad. Wet. Ser. A , 64 : 4 (1961) pp. 362–403 |

[4] | P. Deligne, "Equations différentielles à points singuliers réguliers" , Lect. notes in math. , 163 , Springer (1970) |

[5] | J. Plemelj, "Problems in the sense of Riemann and Klein" , Wiley (1964) |

[6] | V.I. Arnol'd, Yu.S. Il'yashenko, "Ordinary differential equations" , Encycl. Math. Sci. , 1 , Springer (Forthcoming) (Translated from Russian) |

#### Comments

Any second-order equation (1) with three regular singular points can be reduced to the hypergeometric equation. In the case of four regular singular points it can be reduced to Heun's equation [a1], Sect. 15.3, which includes an algebraic form of the Lamé equation. Extensions of the concept to systems of partial differential equations are mentioned in (the editorial comments to) Hypergeometric equation.

#### References

[a1] | H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 3. Automorphic functions , McGraw-Hill (1955) |

**How to Cite This Entry:**

Regular singular point.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Regular_singular_point&oldid=11770