Eigenfunction 2002-06-17 06:24:25 2003-05-07 01:55:51 Definition solution of system Sturm-Liouville % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Consider the Sturm-Liouville system given by \begin{equation} \frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+q(x)y+\lambda r(x)y=0\;\;\;\;\;\;a\leq x\leq b \label{stuff} \end{equation} \begin{equation} a_{1}y(a)+a_{2}y^{\prime}(a)=0,\;\;\; \;\;\;b_{1}y(b)+b_{2}y^{\prime}(b)=0, \label{stuff1} \end{equation} where $a_{i},b_{i}\in \mathbb{R}$ with $i\in \{1,2\}$ and $p(x),q(x),r(x)$ are differentiable functions and $\lambda\in\mathbb{R}$. A non zero solution of the system defined by \eqref{stuff} and \eqref{stuff1} exists in general for a specified $\lambda$. The functions corresponding to that specified $\lambda$ are called eigenfunctions. More generally, if $D$ is some linear differential operator, and $\lambda\in \mathbb{R}$ and $f$ is a function such that $Df=\lambda f$ then we say $f$ is an eigenfunction of $D$ with eigenvalue $\lambda$.