Approximate formulas for moderately small eikonal amplitudes

We consider the eikonal approximation for moderately small scattering amplitudes. To find numerical estimates of these approximations, we derive formulas that contain no Bessel functions and consequently no rapidly oscillating integrands. To obtain these formulas, we study improper integrals of the first kind containing products of the Bessel functions $J_0(z)$. We generalize the expression with four functions $J_0(z)$ and also find expressions for the integrals with the product of five and six Bessel functions. We generalize a known formula for the improper integral with two functions $J_\nu(az)$ to the case with noninteger $\nu$ and complex $a$.