Complete the evaluation of the remaining Meijer G-function limits

Evaluate the required limits of the Meijer G-function representations of the next-to-leading-order Fourier-transform integrals in order to obtain complete closed-form expressions for those integrals.

Background

The paper reduces the next-to-leading-order Fourier transforms governing massive-scalar gravitational scattering to one-dimensional parametric integrals. In an alternative formulation, two of these remaining integrals are represented through limits or derivatives of Meijer G-functions.

Although the Meijer G-function representation provides a formal special-function description, the required limits needed to completely evaluate the integrals are not obtained. Completing these limits would determine whether the generalized special-function forms can yield closed expressions more explicit than the parametric representations used in the main calculation.

References

Indeed, we were not able to take the required limits in order to completely evaluate these integrals.

— Classical gravitational scattering with a massive scalar mediator  (2608.14267 - Biendarra et al., 14 Aug 2026) in Appendix, Section “Alternative forms in terms of Meijer G-functions” (Section “Fourier transforms”)