regular singular point

Let a meromorphic linear ordinary differential equation

y (n)+h 1y (n1)++h ny=0(E) y^{(n)}+h_1 y^{(n-1)}+ \ldots + h_n y = 0\,\,\,\,\,\,\,\,\,\, (E)

defined everywhere in the complex plane (in dimension 1) outside of a finite set of singular points (where some of the meromorphic coefficients have singularity), is given. Equivalently, we can instead of equation (E) of higher order, look at a system

dudz=Au(S) \frac{\mathrm{d}u}{\mathrm{d}z} = A u \,\,\,\,\,\,\,\,\,\,(S)

of meromorphic first order differential equations where AA is the matrix of coefficients and z nz\in\mathbb{C}^n is a column vector.

A singular point z 0z_0 is regular singular if the module of solutions of the equation in a neighborhood of that point the solutions make a module over the algebra of (convergent) meromorphic functions possibly with pole at z 0z_0 which is generated by functions of the form z a(logz) bz^a (log z)^b where aa\in \mathbb{C} and b 0b\in \mathbb{N}_0. If a singular point is not regular it is called irregular singular point.

It can be shown that z 0z_0 is regular singular if a formal Laurent power expansion of a formal solution around z 0z_0 every small sector around z 0z_0 has a branch bounded by |z| N|z|^{-N} for some N0N\geq 0 as z0z\to 0.

Fuchs criterium: (E) has a regular singular point at 00 iff ord(h i)iord(h_i)\geq -i for 1in1\leq i\leq n where ordord is the order function at 00. There is no similar iff criterium for the coefficients of the linear system. However there is a sufficient condition due Savage: (S) has a regular singular point at 00 if )) is a simple singularity of all entries of the connection matrix. This condition is clearly not invariant under the meromorphic changes of coordinates.

This notion goes back to Riemann and Fuchs. Equations whose all singular points are regular singular are called Fuchsian.

  • wikipedia: regular singular point

  • Yu. S. Il’yashenko, Regular singular point, Springer Enc. of Math. 2001

  • V.S. Varadarajan, Linear meromorphic differential equation: a modern point of view, Bull. AMS 33, n. 1, 1996, pdf

  • P. Deligne, Équations différentielles à points singuliers réguliers, Lect. Notes in Math. 163, Springer-Verlag (1970)

Last revised on March 25, 2010 at 21:02:05. See the history of this page for a list of all contributions to it.