Liouville–Neumann series

2025-07-02T20:27:26Z

2001:44C8:6590:3641:A88C:E4FF:FE9A:286D:

In [[mathematics]], the '''Liouville–Neumann series''' is a function series that results from applying the [[resolvent formalism]] to solve [[Fredholm integral equation]]s in [[Fredholm theory]].

==Definition==
The Liouville–Neumann series is defined as
:<math>\phi\left(x\right) = \sum^\infty_{n=0} \lambda^n \phi_n \left(x\right)</math>
which, provided that <math>\lambda</math> is small enough so that the series converges, is the unique [[continuous function|continuous]] solution of the [[Fredholm integral equation]] of the second kind,

{{Equation box 1
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|equation = <math>f(x)= \phi(x) - \lambda \int_a^bK(x,s)\phi(s)\,ds.</math>
|cellpadding= 6
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|border colour = #0073CF
|bgcolor=#F9FFF7}}

If the ''n''th iterated [[Kernel (integral operator)|kernel]] is defined as ''n''−1 nested integrals of ''n'' operator kernels {{mvar|K}},
:<math>K_n\left(x,z\right) = \int\int\cdots\int K\left(x,y_1\right)K\left(y_1,y_2\right) \cdots K\left(y_{n-1}, z\right) dy_1 dy_2 \cdots dy_{n-1}</math>
then
:<math>\phi_n\left(x\right) = \int K_n\left(x,z\right)f\left(z\right)dz</math>
with
:<math>\phi_0\left(x\right) = f\left(x\right)~,</math>
so ''K''<sub>0</sub> may be taken to be {{math|''δ''(''x−z'')}}, the kernel of the [[identity operator]].

The [[Resolvent formalism|resolvent]], also called the "solution kernel" for the integral operator, is then given by a generalization of the [[Geometric series#Generalizations beyond real and complex values|geometric series]],
:<math>R\left(x, z;\lambda\right) = \sum^\infty_{n=0} \lambda^n K_{n} \left(x, z\right),
</math>
where ''K''<sub>0</sub> is again {{math|''δ''(''x−z'')}}.

The solution of the integral equation thus becomes simply
:<math>\phi\left(x\right) = \int R\left( x, z;\lambda\right) f\left(z\right)dz.</math>

Similar methods may be used to solve the [[Volterra integral equation]]s.

==See also==
*[[Neumann series]]

==References==
* Mathews, Jon; Walker, Robert L. (1970), ''Mathematical methods of physics'' (2nd ed.), New York: W. A. Benjamin, {{isbn|0-8053-7002-1}}
* {{Citation |last=Fredholm|first=Erik I. | authorlink = Erik Ivar Fredholm |title=Sur une classe d'equations fonctionnelles|journal=Acta Mathematica|date=1903|volume=27|pages=365–390 |doi=10.1007/bf02421317|url=https://zenodo.org/record/1925972|doi-access=free}}

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[[Category:Fredholm theory]]
[[Category:Series (mathematics)]]
[[Category:Mathematical physics]]

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Liouville–Neumann series