As stated in Morse and Feshbach [4] an example of a second-order differential equation with one regular singular point is

In total for expansions around a single regular singular point, we have twenty-four equivalent solutions, obtained by simply transforming the original equation.

has two regular singular points, at zero and at infinity.

This equation has three regular singular points, at zero, one, and infinity.

Solution of equation (14) for the oscillating part of the velocity is obtained as a power series, using

regular singular point method, in the form [17]

We consider the general Sturm-Liouville problem on (a,[infinity]), assuming that the left endpoint x = a is either (1) regular, or (2) nonoscillatory for all real [Lambda] and a regular singular point. We also assume that x = [infinity] falls in Weyl's limit point case, and generates some continuous spectrum.

Currently SLEDGE is only capable of handling the special case when x = a is a regular singular point of (1.1) which is nonoscillatory for all real [Lambda].

In general, when x = a is not a regular singular point but satisfies the requirement of being nonoscillatory for all real [Lambda], a normalization of the principal solution [Phi] which can be easily implemented numerically is not known.

(2) When x = a is a singular endpoint, the assumption that it is also a regular singular point is needed to implement the normalization (1.13), so it suffices to assume that p, q, and r satisfy the assumptions for a RSP in a small neighborhood of x = a.

Among specific topics are the complex exponential function, two basic equations and their monodromy,

regular singular points and the local Riemann-Hilbert correspondence, the universal group as the pro-algebraic hull of the fundamental group, and beyond local fuschian differential Galois theory.

We note that the Heun operator [H.sup.+.sub.[lambda]] (w, [[partial derivative].sub.w]) has four

regular singular points, w = 0,1,[alpha][beta] and [infinity].

Special attention is paid to the linear ODE, the operational methods for differential systems of equations (substitution method, Cramer's rule etc.) to solve algebraic systems, the Laplace transform and the series method for solution about ordinary and

regular singular points (Froebenius method [1-3]).