Special Functions

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By solving Partial Differential Equations by Separation of Variables, one reduces the PDE in several ODEs. Very common ODEs that originate from PDEs are special functions like

This ODEs that they belong to a greater class of Oridinary Differential Equations, the so called Sturm-Liouville Differential Equations, thus the discussion of special functions will lead us to the Sturm-Liouville theory.

Let us consider a general ODE of the form \begin{align} y''+r(x)y'+q(x)y+ \lambda \rho(x) y=0 \tag{1} \end{align} In analogy to the Integrating Factor method to solve ODEs, we multiply (1) by the Integrating Factor $u=e^{\int r(x) dx}$. \begin{align} u(x)y''+u(x) r(x)y'+u(x) q(x)y+ u(x)\lambda \rho(x) y=0 \end{align} Then we can write the same equation as \begin{align} (u(x)y')'+\underbrace{u(x) q(x)}_{\tilde{q}(x)}y+ \lambda \underbrace{u(x)\rho(x)}_{\tilde{\rho}(x)} y=0 \end{align} with $(u(x)y')'=u(x)y''+u(x) r(x)y'$ since $u'=r \, u$

This is the Sturm-Liouville equation \begin{align} \left [ u(x)\frac{dy}{dx} \right]'+\tilde{q}(x)y+ \lambda \tilde{\rho}(x) y=0\\ \end{align}

Sturm-Liouville equations have the special property, that their eigenfunctions are orthogonal. This key-property allows to expand arbitrary functions from initial conditions into an orthogonal series. In the following a few well known equations are represented in Sturm-Liouville form

  • Simple harmonic equation

\begin{align} (y')'+\omega^2y=0\\ \end{align}

  • Legendre's equation

\begin{align} \left [ (1-x^2)\frac{dy}{dx} \right]'+\ell(\ell+1) y=0\\ \end{align}

  • Bessel's equation

\begin{align} \left [ \xi\frac{dy}{dx} \right]'+ \left( a^2 \xi - \frac{n^2}{\xi} \right) y=0\\ \end{align}