Eigenfunction

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as

for some scalar eigenvalue λ. The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions.

An eigenfunction is a type of eigenvector.

In general, an eigenvector of a linear operator D defined on some vector space is a vector that, when D acts upon it, does not change direction and instead is simply scaled by some scalar value called an eigenvalue. In the special case where D is defined on a function space, the eigenvectors are referred to as eigenfunctions. That is, a function f is an eigenfunction of D if it satisfies the equation

${\displaystyle Df=\lambda f,}$

 

 

 

 

()

where λ is a scalar. The solutions to Equation (1) may also be subject to boundary conditions. Because of the boundary conditions, the possible values of λ are generally limited, for example to a discrete set λ₁, λ₂, ... or to a continuous set over some range. The set of all possible eigenvalues of D is sometimes called its spectrum, which may be discrete, continuous, or a combination of both.

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