### 4.9 Integral Equations

The integral equation under consideration in this context is the Fredholm integral equation of the second
kind, which is commonly given in the form

is the kernel of the integral equation. As the solution appears on both sides of
Equation 4.179, it can be inserted into itself, thus yielding
Thus the function takes the form of a series

where
While
this recursion relation is straightforward, an explicit expression for the depending only on
is favourable. It can be obtained by rewriting the recursion with a focus on the kernels
instead of the functions to read.
Now
the are given as
which
can be generalized to
This
representation is known as iterated kernels. The resulting series is known as a Neumann series or the
resolvent series of the Equation 4.179 and takes the form
Thus
the expression of Equation 4.181 may also be expressed as