Liouville–Neumann series

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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.


The Liouville–Neumann (iterative) 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 K,



so K0 may be taken to be δ(x−z).

The resolvent (or solving kernel for the integral operator) is then given by a schematic analog "geometric series",

where K0 has been taken to be δ(x−z).

The solution of the integral equation thus becomes simply

Similar methods may be used to solve the Volterra equations.

See also[edit]


  • Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin, ISBN 0-8053-7002-1
  • Fredholm, Erik I. (1903), "Sur une classe d'equations fonctionnelles", Acta Mathematica, 27: 365–390, doi:10.1007/bf02421317