eigenintegral

An eigenintegral is a mathematical concept that arises in the study of integral equations. It is analogous to the idea of eigenvalues and eigenvectors in linear algebra. Just as an eigenvector of a linear operator remains unchanged in direction when the operator is applied to it, an eigenfunction of an integral operator remains unchanged in form when the integral operator is applied to it. The eigenintegral is the result of applying the integral operator to such an eigenfunction.

More formally, consider a linear integral operator K, defined by (Kf)(x) = \int_a^b K(x, t) f(t) dt, where K(x, t) is the kernel of the integral equation. If there exists a function \phi(x) and a scalar \lambda such that K\phi = \lambda\phi, then \phi is an eigenfunction of the operator K, and \lambda is its corresponding eigenvalue. In this context, an eigenintegral is the expression \int_a^b K(x, t) \phi(t) dt, which, by definition of an eigenfunction, is equal to \lambda\phi(x).

Eigenintegrals and eigenfunctions are fundamental to solving many types of integral equations, particularly Fredholm integral equations. The set of eigenfunctions often forms a complete basis, allowing for the representation of other functions as a series or integral of these eigenfunctions. This decomposition is crucial for analyzing the behavior of solutions to differential and integral equations that appear in physics, engineering, and other scientific disciplines. The properties of the kernel K(x, t) play a significant role in determining the nature and existence of these eigenfunctions and their corresponding eigenintegrals.