Asymptotics of the contour of the stationary phase and efficient evaluation of the Mellin-Barnes integral for the F_3 structure function

Abstract

A new approximation is proposed for the contour of the stationary phase of the Mellin--Barnes integrals in the case of its finite asymptotic behavior as ${\rm Re} z\to -\infty$. The efficiency of application of the proposed contour and the quadratic approximation to the contour of the stationary phase is compared by the example of the inverse Mellin transform for the structure function $F_3$. It is shown that, although for a small number of terms $N$ in quadrature formulas used to calculate integrals along these contours, the quadratic contour is more efficient, but for $N>20$ the asymptotic stationary phase integration contour gives better accuracy. The case of the $Q^2$-dependence of the $F_3$ structure function is also considered.