Exact Inverse-Separation Series for Multiple Scattering in Two Dimensions

We consider configurations of arbitrary scatterers (s=1,…, N) in two dimensions, such that the circles circumscribing the scatterers do not intersect. As shown previously [V. Twersky, in Electromagnetic Waves, R. E. Langer, Ed. (University of Wisconsin Press, Madison, 1962), pp. 361-389], the solution can be written in terms of the multiple-scattered scattering amplitudes G8, and the G8 are specified by the presumably known farfield isolated scattering amplitudes g8, by a set of integral equations G(g) (which can be converted to algebraic equations involving Hankel functions of the separations b8 t, etc.). Among other applications, the previous paper gave the complete asymptotic series for G(g) in inverse powers of the b's; this was based essentially on Hankel's asymptotic expansion for the Hankel functions Hn. The present paper derives the analogous convergent representation of G(g) based on the exact representation of Hn in terms of Lommel polynomials. For N scatterers, we give the multiple-scattering solution as a series in H0, H1, b-n, and the derivatives of g with respect to angles. For two scatterers, we give a closed form in terms of a differential operator.