Regular singular point

A notion in the theory of ordinary linear differential equations with an independent complex variable.

Regularity as a growth condition for solutions

A point $t_*\in\CC$ is called a regular singular point of the equation $$y^{(n)}+a_1(t)y^{(n-1)}+\cdots+a_{n-1}(t)y'+a_n(t)y=0\label{(1)}$$ or of the system $$\dot z=A(t)z,\quad z\in\CC^n,\ A(t)=\|a_{ij}(t)\|_{i,j=1}^n$$ with coefficients $a_j(\cdot)$, resp., $a_{ij}(t)$ meromorphic at the point $t_*$, if every solution of the equation (resp., the system) increases no faster than polynomially as $t\to t_*$ in any sector. This means that for any proper sector $\{\alpha<\arg (t-t_*)<\beta\}$ with $\beta-\alpha<\pi$ any solution $y_*(t)$ of the equation (resp., any vector solution $z_*(t)$ of the system) is constrained by an inequality of the form $$ |y_*t)|\le C|t-t_*|^{-d},\quad\text{resp.,}\quad \|z_*(t)\|\le C|t-t_*|^{-d},\qquad 0<C,d<+\infty \label{(2)} $$ with suitable constants $C,d$.

Regular singularities constitute a simplest type of singularities of multivalued functions, closely analogous to polar singularities of single-valued functions. There is a simple condition on the coefficients, called the Fuchs condition, which guarantees that the equation (resp., system) has a regular singularity.

Fuchsian condition

1. The $j$th coefficient $a_j(t)$ of the scalar equation (1) has a pole of order $\leqslant j$ at $t=t_*$: $(t-t_*)^j a_j(t)$ extends holomorphically at the point $t_*$ for all $j=1,\dots,n$;
2. The matrix function $A(t)$ has a pole of order 1 (at worst) at the point $t=t_*$: the product $(t-t_*)A(t)$ admits extension as a holomorphic matrix function at the point $t_*$.

The key difference between the equation (1) and the system (2) is the necessity of the Fuchsian condition for the regularity: any equation exhibiting a regular singular point satisfies the Fuchsian condition at this point, whereas a system with a pole of order $\geqslant 2$ may well be regular.

Multidimensional generalization

Apart from ordinary linear equations and systems, the notion of a regular singularity exists also in the theory of (integrable) Pfaffian systems.

References

[Go]	V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen", Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)
[In]	E. L. Ince, "Ordinary Differential Equations", Dover Publications, New York (1944), MR0010757
[D]	P. Deligne, Équations différentielles à points singuliers réguliers, (French) Lecture Notes in Mathematics, Vol. 163. Springer-Verlag, Berlin-New York, 1970 MR0417174
[IY]	Yu. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations. Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008 MR2363178

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)