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

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 $\mathrm{Re}z\to -\infty$. The efficiency of application of the proposed contour and the quadratic approximation to the contour of the stationary phase is compared to 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.

Figure 1

The efficient contours for the MB integral (14). The solid, dotted, and dashed curves indicate the contours $C_{as}$, $C_{st}$, and $C_K$, respectively. Horizontal lines show the asymptotics (19).

Figure 2

The comparison of the efficient contours for the $F_3$ structure function at $x_{\mathtt{B}}=0.1$. The notations are the same as in Fig. 1. The contour ${\overline{C}}_K$ is shown as the dash-dotted curve. The horizontal lines correspond to the asymptotics (33).

Figure 3

The comparison of the behavior of the saddle point $c_0$ and the point $c_{as}$ versus $x_{\mathtt{B}}$ at two fixed values of $Q^2$.

Figure 4

The comparison of the relative accuracy $ϵ(N)$ of the $x_{\mathtt{B}}{F}_3(x_{\mathtt{B}},Q^2)$ reconstruction versus the number of terms $N$ in the sum (23) at $x_{\mathtt{B}}=0.01$ (triangles) and $x_{\mathtt{B}}=0.5$ (squares) using the contour $C_{as}(x_{\mathtt{B}},Q_0^2)$ (open triangles and squares) and $C_{as}(x_{\mathtt{B}},Q^2)$ (full triangles and squares) for $Q_0^2=3{\mathrm{GeV}}^2$ and $Q^2=100{\mathrm{GeV}}^2$.

Figure 5

The comparison of the efficient contours for the MB integral (14) in the region $-s>4$. The notations are the same as in Fig. 1.