Liouville–Neumann series
In mathematics, the Liouville–Neumann series is an infinite series that corresponds to the resolvent formalism technique of solving the Fredholm integral equations in Fredholm theory.Definition
The Liouville–Neumann series is defined as
which, provided that is small enough so that the series converges, is the unique continuous solution of the Fredholm integral equation of the second kind,
If the nth iterated kernel is defined as n−1 nested integrals of n operators,
then
with
so K0 may be taken to be.
The resolvent is then given by a schematic analog "geometric series",
where K0 has been taken to be.
The solution of the integral equation thus becomes simply
Similar methods may be used to solve the Volterra equations.
