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 () and () 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 $\lambda$