Integral Equations As A Generalization of Eigenvalue Equations
Certain homogeneous linear integral equations can be viewed as the continuum limit of eigenvalue equations. Using index notation, an eigenvalue equation can be written as
- ,
where is a matrix, is one of its eigenvectors, and is the associated eigenvalue.
Taking the continuum limit, by replacing the discrete indices and with continuous variables and, gives
- ,
where the sum over has been replaced by an integral over and the matrix and vector have been replaced by the 'kernel' and the eigenfunction . (The limits on the integral are fixed, analogously to the limits on the sum over .) This gives a linear homogeneous Fredholm equation of the second type.
In general, can be a distribution, rather than a function in the strict sense. If the distribution has support only at the point, then the integral equation reduces to a differential eigenfunction equation.
Read more about this topic: Integral Equation
Famous quotes containing the word integral:
“Painting myself for others, I have painted my inward self with colors clearer than my original ones. I have no more made my book than my book has made me—a book consubstantial with its author, concerned with my own self, an integral part of my life; not concerned with some third-hand, extraneous purpose, like all other books.”
—Michel de Montaigne (1533–1592)