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where the sum over has been replaced by an integral over and the matrix and vector have been replaced by the 'kernel' and the eigenfunction .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.
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