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., $a_{ij}(\cdot)$ 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 $C,d$. The point $t_*=\infty$ is regular, if the equation (resp., the system) has a regular singularity at the point $\tau=0$ after the change of the independent variable $t=1/\tau$.

Regular singularities constitute a simplest type of singularities of multivalued functions, closely analogous to polar singularities of single-valued functions.

A singular point (a pole of coefficients) which is not regular, usually referred to as an irregular singularity.

Fuchsian condition

There is a simple condition on the coefficients, called the Fuchs condition, which guarantees that the equation (resp., system) has a regular singularity.

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.

Example: Euler equation, Euler system

The equation $t^n y^{(n)}+c_1t^{n-1}\,y^{(n-1)}+\cdots+c_{n-1}\,ty'+c_n\, y=0$ with constant coefficients $c_1,\dots,c_n\in\CC$ is Fuchsian at the points $t=0,\infty$ (and nonsingular at all other points). The system $t\dot z=A z$ with a constant $n\times n$-matrix $A$ has two Fuchsian singular points at $t=0,\infty$.

Special cases

Any second-order $(n=2)$ equation (1) with three regular singular points on the Riemann sphere $\CC\cup\infty$ can be reduced to the hypergeometric equation. In the case of four regular singular points it can be reduced to Heun's equation [B, 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

Multidimensional generalization

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

Bibliography

[In]	E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944, MR0010757
[H]	P. Hartman,Ordinary differential equations, Classics in Applied Mathematics 38, Corrected reprint of the second (1982) edition, SIAM Publ., Philadelphia, PA, 2002, MR0658490, MR1929104
[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
[B]	H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 3. Automorphic functions , McGraw-Hill (1955)