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# Fredholm integral equation

In mathematics, the Fredholm integral equation introduced by Ivar Fredholm gives rises to a Fredholm operator. From the point of view of functional analysis it therefore has a well-understood abstract eigenvalue theory. In this case that is supported by a computational theory, including the Fredholm determinants .

An inhomogeneous Fredholm equation of the first kind is written:

$g(t)=\int_a^b K(t,s)f(s)\,ds$

and the problem is, given the continuous kernel function K(t,s), and the function g(t), to find the function f(s).

An inhomogeneous Fredholm equation of the second kind is essentially a form of the eigenvalue problem for the above equation:

$f(t)=\lambda\int_a^bK(t,s)f(s)\,ds + g(t)$

and the problem is again, given the kernel K(t,s), and the function g(t), find the function f(s). The kernel K is a compact operator (to show this one relies on equicontinuity). It therefore has a spectral theory that can be understood in terms of a discrete spectrum of eigenvalues that tend to 0. This underlies the theory of the equation.