# Sturm-Liouville boundary value problems

• Martin Braun
Chapter
Part of the Texts in Applied Mathematics book series (TAM, volume 11)

## Abstract

In Section 5.5 we described the remarkable result that an arbitrary piecewise differentiable function f(x) could be expanded in either a pure sine series of the form
$$f\left( x \right) = \sum\limits_{n = 1}^\infty{{b_n}\sin \frac{{n\pi x}}{l}}$$
(1)
or a pure cosine series of the form
$$f\left( x \right) = \frac{{{a^0}}}{2}\sum\limits_{n = 1}^\infty{{a_n}\cos \frac{{n\pi x}}{l}}$$
(2)
on the interval 0 < x < l. We were led to the trigonometric functions appearing in the series (1) and (2) by considering the 2 point boundary value problems
$$y'' + \lambda y = 0,y\left( 0 \right) = 0,y\left( l \right) = 0,$$
(3)
and
$$y'' + \lambda y = 0,y'\left( 0 \right) = 0,y'\left( l \right) = 0.$$
(4)
Recall that Equations (3) and (4) have nontrivial solutions
$${y_n}\left( x \right) = c\sin \frac{{n\pi x}}{l}and{\kern 1pt} {y_n}\left( x \right) = c\cos \frac{{n\pi x}}{l},$$
respectively, only if λ = λn = n2π2/l2. These special values of λ were called eigenvalues, and the corresponding solutions were called eigenfunctions.

## Keywords

Nontrivial Solution Product Space Orthogonal Basis Hermitian Operator Laguerre Polynomial
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