Definition:Eigenfunction
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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