Regular singular point

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In mathematics, in the theory of ordinary differential equations in the complex plane C, the points of C are classified into ordinary points, at which the equation's coefficients are analytic functions, and singular points, at which some coefficient has a singularity. Then amongst singular points, an important distinction is made between a regular singular point, where there are meromorphic function solutions in Laurent series, and an irregular singular point, where the full solution set requires functions with an essential singularity. This distinction occurs, for example, between the hypergeometric equation, with three regular singular points, and the Bessel equation which is in a sense a limiting case, but where the analytic properties are substantially different.

More precisely, suppose a linear, nth-order equation written

Lf = Σ p_i (z)f⁽ⁱ⁾(z) = 0

with p_i (z) meromorphic functions and p_n (z) = 1; we can make the latter condition hold by dividing through, though naturally this may introduce singular points to consider. Also, properly, the equation should be studied on the Riemann sphere, so that the point at infinity is a possible singular point; as usual, a Möbius transformation may be applied to move ∞ into the finite part of the complex plane if required, and in any case it plays no special part in the theory, so long as its presence is not forgotten.

Then the Frobenius method based on the indicial equation may be applied to find possible solutions that are Laurent series near any given a in the complex plane. This presents no difficulty for a an ordinary point (Lazarus Fuchs 1866). When a is a regular singular point, which by definition means that

p_{n − i} (z)

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. To be strictly accurate, solutions will be a Laurent series at a multiplied by a power

(z − a)^r

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.

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.

[edit] References

• E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (1935)
• E. L. Ince, Ordinary Differential Equations, Dover Publications (1944)