Eigenfunction
In mathematics, an
eigenfunction of a linear operator, A, defined on some function space, is any non-zero function
f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has for some scalar, λ, the corresponding eigenvalue. The solution of the differential eigenvalue problem also depends on any boundary conditions required of
f . In each case there are only certain eigenvalues
λ =
λn that admit a corresponding solution for
f =
fn when combined with the boundary conditions. Eigenfunctions are used to analyze A. For example,
fk =
ekx is an eigenfunction for the differential operator for any value of k, with corresponding eigenvalue
λ =
k2 −
k. If boundary conditions are applied to this system, then only certain values of
k =
kn satisfy the boundary conditions, generating corresponding discrete eigenvalues.