The O(α$^{2}$) initial state QED corrections to e$^{+}$ e$^{-}$ $\rightarrow$ $\gamma$$^{*}$ / Z$_{0}^{*}$

We calculate the complete O($\alpha$$^{2}$) initial state radiation corrections to e$^{+}$ e$^{-}$ annihilation into a neutral vector boson in a direct analytic computation without any approximation. The corrections are represented in terms of iterated incomplete (elliptic) integrals over alphabets of square-root valued letters. Performing the limit s >> m$_{e}^{2}$, we find discrepancies with the earlier results of Ref. [1] and confirm results obtained in Ref. [2] where the effective method of massive operator matrix elements has been used, which works for all but the power corrections in m$_{e}^{2}$/s. In this way, we also confirm the validity of the factorization of massive partons in the Drell-Yan process to O($\alpha$$^{2}$). We add non logarithmic terms at O($\alpha$$^{2}$) which have not been considered in previous calculations. The final results in the limit s >> m$_{e}^{2}$ can be given in terms of Nielsen integrals