More precisely, consider an ordinary linear differential equation of n-th order
with pi (z) meromorphic functions. One can assume that
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 is 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 (z − a)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
has a pole of order at most i at a, the Frobenius method also can be made to work and provide n independent solutions near a.
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 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.
One distinguishes the following cases:
Listed below are several examples from ordinary differential equations from mathematical physics that have singular points and known solutions.
Dividing this equation by x2 gives:
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.
To see what happens when x → ∞ one has to use Möbius transformation, for example x = 1 / (w - b). After performing the algebra:
Now p1(w) = 1 / (w - b) has a pole of first order at w = b. And p0(w) has a pole of fourth order at w = b. Thus this equation has an irregular singularity w = b corresponding to x at ∞.
The solutions of this differential equation are Bessel functions.
Opening the square bracket gives:
And dividing by (1 - x2):
This differential equation has regular singular points at -1, +1, and ∞.
for harmonic oscillator. In this case the potential energy V(x) is:
This leads to the following ordinary second order differential equation is:
This differential equation has an irregular singularity at ∞. Its solutions are Hermite polynomials.
Dividing both sides by z (1 - z) gives:
This differential equation has regular singular points at 0, 1 and ∞. The solutions is the hypergeometric series.
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