The Mathematica Book
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The triply differential cross-section for the inelastic Compton scattering of photons by K-shell bound electrons is expressed in closed form in terms of four high transcendental Lauricella functions FD of four variables and six parameters, all complex. In order to obtain the doubly differential cross-section, a numerical integration over the directions of the final electron is needed. The aim of this paper is to present a method for both evaluation and integration over the solid angle of Lauricella functions, valid for any photon energy and target's atomic number. While all the researchers seem to avoid even the evaluation of these functions, we present a new quadrature method, based on the analysis of the pathological behavior of the integrand near origin. Also, the solid angle integration process is achieved and discussed. Keeping the full form of the Lauricella functions allowed us to include in the doubly differential cross-section the exact dependence on the angles describing the directions of the final electron and photon. The accuracy of evaluating the Lauricella functions was checked by using recurrence relations between them, and a fairly good precision was observed for any physical parameters and variables occurring in two photon bound-free electron transitions. We mention that our analytical approach of the triply differential cross-section allows the numerical calculation of the doubly differential one in seconds using an average personal computer. Also, our results based on these evaluations display a good agreement with direct numerical calculations of the S-matrix element performed by Bergstrom et al [Phys. Rev. A 48, 1134 (1993)] on Cray computers.