Regular singular point
2020 Mathematics Subject Classification: Primary: 34M03,34M35 Secondary: 32Sxx [MSN][ZBL]
A notion in the theory of ordinary linear differential equations with an independent complex variable. A point in the plane of the independent variable is regular singular, if solutions of the equation loose analyticity, but exhibit at most polynomial growth at this point.
Regularity as a growth condition for solutions
A point $t_*\in\CC$ is called a regular singular^{[1]} 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^{[2]} 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.
- 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 $j=1,\dots,n$;
- 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.
Notes
- ↑ The construct "regular singularity", which is an oxymoron, is too firmly rooted to be replaced by terms like "moderate" or "tame" singularity as was suggested in [IY]. The regular singular point should not be confused with a regular (nonsingular) point at which the coefficients $a_j(t)$, resp., $a_{jk}(t)$, are holomorphic.
- ↑ In particular, the coefficients should be holomorphic in a punctured neighborhood of $t_*$ and at worst a finite order pole at it.
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) |
Regular singular point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_singular_point&oldid=55915