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Liouville-Neumann series ：ウィキペディア英語版

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
:

\phi\left(x\right) = \sum^\infty_ \lambda^n \phi_n \left(x\right)

which is a unique, continuous solution of a Fredholm integral equation of the second kind:
:

f(t)= \phi(t) - \lambda \int_a^bK(t,s)\phi(s)\,ds.

If the ''n''th iterated kernel is defined as
:

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_, z\right) dy_1 dy_2 \cdots dy_

then
:

\phi_n\left(x\right) = \int K_n\left(x,z\right)f\left(z\right)dz

with
:

\phi_0\left(x\right) = f\left(x\right).

The resolvent or solving kernel is given by
:

K\left(x, z;\lambda\right) = \sum^\infty_ \lambda^n K_ \left(x, z\right).

The solution of the integral equation becomes
:

\phi\left(x\right) = \int K \left( x, z;\lambda\right) f\left(z\right)dz.

Similar methods may be used to solve the Volterra equations.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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