The Inverse Mellin Transform Method for the Fragmentation Functions Based on the Efficient Approximations for the Contour of the Stationary Phase

We apply the Mellin transformation method to the recovery form of the fragmentation function using two efficient approximations for the contour of the stationary phase. The first contour is of parabolic form and runs from the saddle point. A new approach is proposed for the contour of the stationary phase of the Mellin–Barnes integrals in the case of their finite asymptotic behavior. The fragmentation functions have a more complicated parametrization in comparison with the structure functions and parton distributions, and this causes the peculiarities in applying the second method for constructing an effective contour. We compare the efficiency of application of these contours for the inverse Mellin transform of the fragmentation functions. It is shown that both methods make it possible to achieve high accuracy in the recovery of the fragmentation function by using a small number of terms in the quadrature formula