TABLE OF CONTENTS In the last section we derived an inhomogeneous integral equation to represent the scattered field, u s( 
), as a function of the object, o( ). This equation can t be solved directly, but a solution can be written using either of the two approximations to be described here. These approximations, the Born and the Rytov, are valid under different conditions but the form of the resulting solutions is quite similar. These approximations are the basis of the Fourier Diffraction Theorem.
Mathematically speaking, (37) is a Fredholm equation of the second kind. A number of mathematicians have presented works describing the solution of scattering integrals [Hoc73], [Col83] which should be consulted for the theory behind the approximations we will present.
The first Born approximation is the simpler of the two approaches. Recall that the total field, u( ), is expressed as the sum of the incident field, u 0( ), and a small perturbation, u s ( ), or
The integral of (37) is now written as
but if the scattered field, u s ( ), is small compared to u 0 ( ) the effects of the second integral can be ignored to arrive at the approximation
An even better estimate can be found by substituting 
in (40) to find
In general, the ith-order Born field can be written
An alternate representation is possible if we write
where
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