# Sturm-Liouville problems

Sturm-Liouville problems

The basic Sturm-Liouville problem is

-(p)+qy=λy

∂

∂x

′

y

with boundary conditions

α 1 α 2 ′ y | = | 0 |

β 1 β 2 ′ y | = | 0 |

If , , and are continuous on and is never zero in , then the problem is called regular. Otherwise, it is singular. A regular Sturm-Liouville problem is guaranteed to have a complete, orthonormal set of eigenfunctions, while a singular problem might not.

p

′

p

q

[a,b]

p

[a,b]

In the most basic example, and , so the eigenfunctions are linear combinations of trigonometric functions; this yields Fourier's theory. The purpose of this notebook is to illustrate the existence of other examples.

p(x)=1

q(x)===0

α

2

β

2

### Assumptions

Assumptions

## Example 1

Example 1

## Example II

Example II

## Example III - Airy functions

Example III - Airy functions

## Example IV - Bessel functions and weighted inner products

Example IV - Bessel functions and weighted inner products