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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 the growth of solutions is bounded (in any small sector) by an algebraic function, and an irregular singular point, where the full solution set requires functions with higher growth rates. 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.
## Formal definitions

## Examples for second order differential equations

In this case the equation above is reduced to:### Bessel differential equation

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

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

One encounters this ordinary second order differential equation in solution of one dimensional time independent Schrodinger equation ### Hypergeometric equation

Is defined as## References

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

- $$

with p_{i} (z) meromorphic functions. One can assume that

- $p\_n(z)\; =\; 1\; ,$

If this is not the case the equation above has to be divided by p_{n}(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

- $p\_\{n-i\}(z),$

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.

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.

$f\text{'}\text{'}(x)\; +\; p\_1(x)\; f\text{'}(x)\; +\; p\_0(x)\; f(x)\; =\; 0,$.

One distinguishes the following cases:

- Point a is ordinary point when functions p
_{1}(x) and p_{0}(x) are analytic at x = a. - Point a is regular singular point if p
_{1}(x) has a pole of order 1 at x = a or p_{0}has a pole of order up to 2 at x = a. - Otherwise point a is irregular singular point.

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

- $x^2\; frac\{d^2\; f\}\{dx^2\}\; +\; x\; frac\{df\}\{dx\}\; +\; (x^2\; -\; alpha^2)f\; =\; 0$

for an arbitrary real or complex number α (the order of the Bessel function). The most common and important special case is where α is an integer n.

Dividing this equation by x^{2} gives:

- $frac\{d^2\; f\}\{dx^2\}\; +\; frac\{1\}\; \{x\}\; frac\{df\}\{dx\}\; +\; left\; (1\; -\; frac\; \{alpha^2\}\; \{x^2\}\; right\; )f\; =\; 0$

In this case p_{1}(x) = 1 / x has a pole of first order at x = 0.
When α ≠ 0 p_{0}(x) = (1 - α^{2} / x^{2}) 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:

- $frac\; \{d^2\; f\}\; \{d\; w^2\}\; +\; frac\; \{1\}\; \{w-b\}\; frac\; \{df\}\; \{dw\}\; +$

Now p_{1}(w) = 1 / (w - b)
has a pole of first order at w = b.
And p_{0}(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.

- $\{d\; over\; dx\}\; left[(1-x^2)\; \{d\; over\; dx\}\; f\; right]\; +\; n(n+1)f\; =\; 0.$

Opening the square bracket gives:

- $(1-x^2)\{d^2\; f\; over\; dx^2\}\; -2x\; \{df\; over\; dx\; \}\; +\; n(n+1)y\; =\; 0.$

And dividing by (1 - x^{2}):

- $\{d^2\; f\; over\; dx^2\}\; -\; \{2x\; over\; (1-x^2)\}\; \{df\; over\; dx\; \}\; +\; \{n(n+1)\; over\; (1-x^2)\}\; f\; =\; 0.$

This differential equation has regular singular points at -1, +1, and ∞.

- $Epsi\; =\; -frac\{hbar^2\}\{2m\}\; frac\; \{d^2\; psi\}\; \{d^2\; x\}\; +\; V(x)psi$

for harmonic oscillator. In this case the potential energy V(x) is:

- $displaystyle\; V(x)\; =\; frac\{1\}\{2\}\; m\; omega^2\; x^2$.

This leads to the following ordinary second order differential equation is:

- $$

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

- $z(1-z)frac\; \{d^2f\}\{dz^2\}\; +$

Dividing both sides by z (1 - z) gives:

- $frac\; \{d^2f\}\{dz^2\}\; +$

This differential equation has regular singular points at 0, 1 and ∞. The solutions is the hypergeometric series.

- E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (1935)
- A. R. Forsyth Theory of Differential Equations Vol. IV: Ordinary Linear Equations (Cambridge University Press, 1906)
- E. Goursat A Course in Mathematical Analysis, Volume II, Part II: Differential Equations p. 128-ff. (Ginn & co., Boston, 1917)
- E. L. Ince, Ordinary Differential Equations, Dover Publications (1944)
- T. M. MacRobert Functions of a Complex Variable p. 243 (MacMillan, London, 1917)
- E. T. Whittaker and G. N. Watson A course of modern analysis p. 188-ff. (Cambridge University Press, 1915)

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Last updated on Thursday August 28, 2008 at 14:09:28 PDT (GMT -0700)

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