  Perturbative solutions for unpolarized QED parton distribution and
fragmentation functions are presented explicitly in the next-to-leading
logarithmic approximation. The scheme of iterative solution of QED evolution
equations is described in detail. Terms up to $\mathcal{O}(\alpha^3L^2)$ are
calculated analytically, where $L=\ln(\mu_F^2/m_e^2)$ is the large logarithm
which depends on the factorization energy scale $\mu_F\gg m_e$. The results are
process independent and relevant for future high-precision experiments.
