Regular singular point#Hermite differential equation

More precisely, consider an ordinary linear differential equation of {{mvar|n}}-th order

$$f^\{(n)\}(z)\; +\; \backslash sum\_\{i=0\}^\{n-1\}\; p\_i(z)\; f^\{(i)\}\; (z)\; =\; 0$$

with {{math|p_i(z)}} meromorphic functions.

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 {{math|(z − a)^r}} near any given {{mvar|a}} in the complex plane where {{mvar|r}} need not be an integer; this function may exist, therefore, only thanks to a branch cut extending out from {{mvar|a}}, or on a Riemann surface of some punctured disc around {{mvar|a}}. This presents no difficulty for {{mvar|a}} an ordinary point (Lazarus Fuchs 1866). When {{mvar|a}} is a regular singular point, which by definition means that

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

has a pole of order at most {{mvar|i}} at {{mvar|a}}, the Frobenius method also can be made to work and provide {{mvar|n}} independent solutions near {{mvar|a}}.

Otherwise the point {{mvar|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 ({{harvtxt|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 {{var|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{{anchor|Fuchsian differential equation}} ordinary differential equation.

In this case the equation above is reduced to:

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

One distinguishes the following cases:

• Point {{mvar|a}} is an ordinary point when functions {{math|p₁(x)}} and {{math|p₀(x)}} are analytic at {{math|1=x = a}}.
• Point {{mvar|a}} is a regular singular point if {{math|p₁(x)}} has a pole up to order 1 at {{math|1=x = a}} and {{math|p₀}} has a pole of order up to 2 at {{math|1=x = a}}.
• Otherwise point {{mvar|a}} is an irregular singular point.

We can check whether there is an irregular singular point at infinity by using the substitution $w\; =\; 1/x$ and the relations:

$$\backslash frac\{df\}\{dx\}=-w^2\backslash frac\{df\}\{dw\}$$

$$\backslash frac\{d^2f\}\{dx^2\}=w^4\backslash frac\{d^2f\}\{dw^2\}+2w^3\backslash frac\{df\}\{dw\}$$

We can thus transform the equation to an equation in {{mvar|w}}, and check what happens at {{math|1=w = 0}}. If $p\_1(x)$ and $p\_2(x)$ are quotients of polynomials, then there will be an irregular singular point at infinite x unless the polynomial in the denominator of $p\_1(x)$ is of degree at least one more than the degree of its numerator and the denominator of $p\_2(x)$ is of degree at least two more than the degree of its numerator.

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:

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

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

Dividing this equation by x² gives:

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

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

To see what happens when {{math|x → ∞}} one has to use a Möbius transformation, for example $x\; =\; 1\; /\; w$. After performing the algebra:

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

\left[ \frac{1}{w^4} - \frac{\alpha ^2}{w^2} \right ] f= 0

Now at {{nowrap|$w\; =\; 0$,}}

$$p\_1(w)\; =\; \backslash frac\{1\}\{w\}$$

has a pole of first order, but

$$p\_0(w)\; =\; \backslash frac\; \{1\}\; \{w^4\}\; -\; \backslash frac\; \{\backslash alpha\; ^2\}\; \{w^2\}$$

has a pole of fourth order. Thus, this equation has an irregular singularity at $w\; =\; 0$ corresponding to x at ∞.

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

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

Opening the square bracket gives:

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

And dividing by {{math|(1 − x²)}}:

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

This differential equation has regular singular points at ±1 and ∞.

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

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

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

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

This leads to the following ordinary second order differential equation:

$$\backslash frac\{d^2\; f\}\{dx^2\}\; -\; 2\; x\; \backslash frac\{df\}\{dx\}\; +\; \backslash lambda\; f\; =\; 0.$$

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

The equation may be defined as

$$z(1-z)\backslash frac\; \{d^2f\}\{dz^2\}\; +\; \backslash left[c-(a+b+1)z\; \backslash right]\; \backslash frac\; \{df\}\{dz\}\; -\; abf\; =\; 0.$$

Dividing both sides by {{math|z(1 − z)}} gives:

$$\backslash frac\; \{d^2f\}\{dz^2\}\; +\; \backslash frac\{c-(a+b+1)z\; \}\; \{z(1-z)\}\; \backslash frac\; \{df\}\{dz\}\; -\; \backslash frac\; \{ab\}\; \{z(1-z)\}\; f\; =\; 0.$$

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

• {{cite book |last=Arscott |first=F.M. |author-link=Felix Arscott |date=1995 |chapter=Heun's Equation |editor=A. Ronveaux |title=Heun's Differential Equations |publisher=Oxford University Press |isbn=0198596952 |url={{GBurl|5p65FD8caCgC|p=74}} |page=74}}
• {{cite book | last1=Coddington | first1=Earl A. | last2=Levinson | first2=Norman | title=Theory of Ordinary Differential Equations | url=https://archive.org/details/theoryofordinary0000unse | url-access=registration | publisher=McGraw-Hill | location=New York | year=1955}}
• E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (1935)
• {{SpringerEOM|title=Fuchsian equation|first=M. V.|last=Fedoryuk}}
• A. R. Forsyth [https://archive.org/details/theoryofdifferen04forsuoft Theory of Differential Equations Vol. IV: Ordinary Linear Equations] (Cambridge University Press, 1906)
• Édouard Goursat, [http://name.umdl.umich.edu/ABA9351.0003.001 A Course in Mathematical Analysis, Volume II, Part II: Differential Equations] pp. 128−ff. (Ginn & co., Boston, 1917)
• E. L. Ince, Ordinary Differential Equations, Dover Publications (1944)
• {{SpringerEOM|title=Regular singular point|first=Yu. S.|last= Il'yashenko}}
• T. M. MacRobert [https://archive.org/details/functionsofcompl00macruoft Functions of a Complex Variable] p. 243 (MacMillan, London, 1917)
• {{cite book | last = Teschl | first = Gerald | authorlink=Gerald Teschl | title = Ordinary Differential Equations and Dynamical Systems | publisher=American Mathematical Society | place = Providence | year = 2012 | isbn = 978-0-8218-8328-0 | url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
• E. T. Whittaker and G. N. Watson A Course of Modern Analysis pp. 188−ff. (Cambridge University Press, 1915)

Category:Ordinary differential equations

Category:Complex analysis