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 hasfor 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 (n = 1, 2, 3, ...) that admit a corresponding solution for  f  =  fn  (with each  fn  belonging to the eigenvalue λn) when combined with the boundary conditions. Eigenfunctions are used to analyze A.For example,  fk (x) = ekx is an eigenfunction for the differential operatorfor any value of k, with corresponding eigenvalue λ = k2 − k. If boundary conditions are applied to this system (e.g.,  f  = 0 at two physical locations in space), then only certain values of k = kn satisfy the boundary conditions, generating corresponding discrete eigenvalues .Specifically, in the study of signals and systems, the eigenfunction of a system is the signal  f (t) which when input into the system, produces a response y(t) = λ f (t) with the complex constant λ.