An exact far-field inversion for the Born approximation and plane wave decompositions for Hankel functions

We consider the Born approximation (representative for first-order approximations) of the scattering problem for the scalar Helroholtz equation with a fixed real-valued free-space wavenumber and a complex-valued compactly supported potential. The boundary condition is the Sommerfeld radiation condition. We derive an exact series-integral representation of the potential from the Fourier coefficients of its far-field pattern, suitable for discussion of the connected stability problem. Furthermore we stress the connection between this representation and some plane wave decompositions for Hankel functions. Without loss of generality we restrict ourselves to the case of two space dimensions.