Special Functions - Main Theorem

The Sturm-Liouville theory provides the unified framework underlying all classical special functions, explaining their orthogonality, completeness, and appearance in eigenvalue problems.

TheoremSturm-Liouville Theorem

Consider the Sturm-Liouville eigenvalue problem:

$\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right] + [q(x) + \lambda w(x)]y = 0$

on interval $[a,b]$ with boundary conditions (e.g., $y(a) = y(b) = 0$ ) where $p, p', q, w$ are continuous and $p(x) > 0$ , $w(x) > 0$ (weight function).

Properties:

All eigenvalues $\lambda_n$ are real
Eigenfunctions $y_n(x)$ corresponding to distinct eigenvalues are orthogonal: $\int_a^b w(x)y_m(x)y_n(x)dx = 0 \quad (m \neq n)$
The eigenvalues form an infinite sequence $\lambda_1 < \lambda_2 < \lambda_3 < \cdots$ with $\lambda_n \to \infty$
The eigenfunctions form a complete basis for $L^2([a,b],w)$

All classical special functions arise as eigenfunctions of Sturm-Liouville problems:

ExampleLegendre Polynomials

Legendre's equation:

$\frac{d}{dx}\left[(1-x^2)\frac{dy}{dx}\right] + \lambda y = 0$

is Sturm-Liouville with $p(x) = 1-x^2$ , $q(x) = 0$ , $w(x) = 1$ . Requiring regularity at $x = \pm 1$ gives eigenvalues $\lambda_n = n(n+1)$ with eigenfunctions $y_n = P_n(x)$ . Orthogonality:

$\int_{-1}^1 P_m(x)P_n(x)dx = \frac{2}{2n+1}\delta_{mn}$

ExampleBessel Functions

Bessel's equation in Sturm-Liouville form:

$\frac{d}{dr}\left[r\frac{dR}{dr}\right] + \left[k^2 r - \frac{n^2}{r}\right]R = 0$

with $p(r) = r$ , $q(r) = -n^2/r$ , $w(r) = r$ , $\lambda = k^2$ . Boundary condition $R(a) = 0$ gives eigenvalues $k_m = \alpha_{nm}/a$ where $J_n(\alpha_{nm}) = 0$ . Orthogonality:

$\int_0^a rJ_n(k_m r)J_n(k_\ell r)dr = \frac{a^2}{2}[J_{n+1}(k_m a)]^2\delta_{m\ell}$

TheoremCompleteness and Expansion Theorem

Any function $f(x) \in L^2([a,b],w)$ can be expanded in eigenfunctions:

$f(x) = \sum_{n=1}^{\infty} c_n y_n(x)$

where:

$c_n = \frac{\int_a^b w(x)f(x)y_n(x)dx}{\int_a^b w(x)y_n^2(x)dx}$

The series converges in the $L^2$ sense, and Parseval's theorem holds:

$\int_a^b w(x)|f(x)|^2dx = \sum_{n=1}^{\infty}|c_n|^2\int_a^b w(x)y_n^2(x)dx$

ExampleFourier-Bessel Series

For a function $f(r)$ on $[0,a]$ , the Fourier-Bessel expansion is:

$f(r) = \sum_{m=1}^{\infty} A_m J_n(\alpha_{nm}r/a)$

where:

$A_m = \frac{2}{a^2[J_{n+1}(\alpha_{nm})]^2}\int_0^a rf(r)J_n(\alpha_{nm}r/a)dr$

This is essential for solving PDEs in cylindrical domains.

TheoremComparison Theorem

Between consecutive zeros $x_1$ and $x_2$ of eigenfunction $y_n(x)$ , there lies at least one zero of $y_{n+1}(x)$ . As $n$ increases, eigenfunctions oscillate more rapidly, related to higher energy states in quantum mechanics.

RemarkGreen's Function Expansion

The Green's function for operator $\mathcal{L}y = -\frac{d}{dx}[p\frac{dy}{dx}] + qy$ expands as:

$G(x,x') = \sum_{n=1}^{\infty}\frac{y_n(x)y_n(x')}{\lambda_n}$

where $y_n$ are normalized eigenfunctions with eigenvalues $\lambda_n$ . This provides the solution to inhomogeneous problems:

$\mathcal{L}u = f \implies u(x) = \int_a^b G(x,x')f(x')w(x')dx'$

The Sturm-Liouville framework unifies the theory of special functions, revealing that their orthogonality and completeness are not coincidental but arise from the self-adjoint nature of the differential operators they satisfy.