eigenfunction

eigenfunction [′ī·gən‚fəŋk·shən]

(mathematics)

Also known as characteristic function.

An eigenvector for a linear operator on a vector space whose vectors are functions. Also known as proper function.

A solution to the Sturm-Liouville partial differential equation.

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Eigenfunction

One of the solutions of an eigenvalue equation. A parameter-dependent equation that possesses nonvanishing solutions only for particular values (eigenvalues) of the parameter is an eigenvalue equation, the associated solutions being the eigenfunctions (sometimes eigenvectors). In older usage the terms characteristic equation and characteristic values (functions) are common. Eigenvalue equations appear in many contexts, including the solution of systems of linear algebraic equations (matrix equations), differential or partial differential equations, and integral equations. The importance of eigenfunctions and eigenvalues in applied mathematics results from the widespread applicability of linear equations as exact or approximate descriptions of physical systems. However, the most fundamental application of these concepts is in quantum mechanics where they enter into the definition and physical interpretation of the theory. Only linear eigenvalue equations will be discussed. See Eigenvalue (quantum mechanics), Energy level (quantum mechanics), Quantum mechanics