Regular singular point

More precisely, consider an ordinary linear differential equation of n-th order

If this is not the case the equation above has to be divided by pn(x). This may introduce singular points to consider.

The equation should be studied on the Riemann sphere to include the point at infinity as a possible singular point. A Möbius transformation may be applied to move ∞ into the finite part of the complex plane if required, see example on Bessel differential equation below.

Then the Frobenius method based on the indicial equation may be applied to find possible solutions that are power series times complex powers (za)r near any given a in the complex plane where r need not be an integer; this function may exist, therefore, only thanks to a branch cut extending out from a, or on a Riemann surface of some punctured disc around a. This presents no difficulty for a an ordinary point (Lazarus Fuchs 1866). When a is a regular singular point, which by definition means that

Otherwise the point a is an irregular singularity. In that case the monodromy group relating solutions by analytic continuation has less to say in general, and the solutions are harder to study, except in terms of their asymptotic expansions. The irregularity of an irregular singularity is measured by the Poincaré rank (Arscott (1995)).

The regularity condition is a kind of Newton polygon condition, in the sense that the allowed poles are in a region, when plotted against i, bounded by a line at 45° to the axes.

An ordinary differential equation whose only singular points, including the point at infinity, are regular singular points is called a Fuchsian ordinary differential equation.

Listed below are several examples from ordinary differential equations from mathematical physics that have singular points and known solutions.

This is an ordinary differential equation of second order. It is found in the solution to Laplace's equation in cylindrical coordinates:

In this case p1(x) = 1/x has a pole of first order at x = 0. When α ≠ 0, p0(x) = (1 − α2/x2) has a pole of second order at x = 0. Thus this equation has a regular singularity at 0.

This is an ordinary differential equation of second order. It is found in the solution of Laplace's equation in spherical coordinates:

One encounters this ordinary second order differential equation in solving the one-dimensional time independent Schrödinger equation

This leads to the following ordinary second order differential equation:

This differential equation has an irregular singularity at ∞. Its solutions are Hermite polynomials.

This differential equation has regular singular points at 0, 1 and ∞. A solution is the hypergeometric function.