Exact and approximate solutions for the electron nonsinglet structure function in QED

Two exact, valid up infinite perturbative order, numerical solutions of the Lipatov equation for the nonsinglet electron struclture function in the QED are presented. One of them is of the Monte Carlo type and another is based on the numerical inversion of the Mellin transform. They agree numerically to a very high precision (better than 0.05%). Within the leading logarithmic approximation, the exact solution is compared with the perturbative second and third order exponentiated solutions. It is shown that the perturbative second order solution inspired by the Yennie-Frautschi-Suura (exclusive) exponentiation is much closer to the exact solution than the other ones. New compact analytical formula for the third order exponential solution is given. It is shown to be in perfect numerical agreement with the infinite order solution of the Monte Carlo and Mellin type. (orig.)