Approximate formulas for moderately small eikonal amplitudes

TL;DR

This paper derives simplified approximate formulas for moderately small eikonal scattering amplitudes, eliminating oscillatory integrals involving Bessel functions to facilitate numerical calculations.

Contribution

The authors develop new formulas that replace Bessel function integrals with more manageable expressions, extending known results to products of up to six Bessel functions and non-integer orders.

Findings

Formulas without Bessel functions for numerical estimation

Generalization to products of up to six Bessel functions

Extension of integrals to non-integer order Bessel functions

Abstract

The eikonal approximation for moderately small scattering amplitudes is considered. With the purpose of using for their numerical estimations, the formulas are derived which contain no Bessel functions, and, hence, no rapidly oscillating integrands. To obtain these formulas, the improper integrals of the first kind which contain products of the Bessel functions J_0(z) are studied. The expression with four functions J_0(z) is generalized. The expressions for the integrals with the product of five and six Bessel functions J_0(z) are also found. The known formula for the improper integral with two functions J_nu(z) is generalized for non-integer nu.