Definition:Eigenfunction

Definition

An eigenfunction is a non-trivial solution to a differential equation subject to boundary conditions involving a parameter, for certain values of that parameter.

Eigenvalue

Let $F$ be an eigenfunction to a differential equation.

The parameter which so defines $F$ is referred to as an eigenvalue of $F$.

Examples

SHM Equation

Consider the differential equation describing simple harmonic motion (SHM):

$\dfrac {\d^2 y} {\d x^2} + \lambda y = 0$

subject to the boundary conditions:

$\map y 0 = 0$

$\map y \pi = 0$

This has an eigenvalue $m^2$ with eigenfunction $\sin m x$ for all non-zero integer $m$.

Also see

• Results about eigenfunctions can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): eigenfunction
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): eigenfunction