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### Description

At $e^+ \, e^-$ colliders the QED--initial state radiation forms a large part of the radiative corrections. Their

precise and fast evaluation is an essential asset for the experiments at LEP, the ILC and the FCC-ee, operating at

high luminosity. A long standing problem in the analytic understanding of the $\mathcal{O}(\alpha^2)$ initial state

radiation is the observed discrepancy between the calculation of Berends et al. (1988) in the limit $m_e^2 \ll s$ and

the result of the effective calculation using massive operator matrix elements by Blümlein et al. (2011) aiming

directly for this limit. In order to resolve this important issue we recalculated this process by integrating directly

over the phase space without any approximation. For parts of the corrections we find exact solutions of the cross section in

terms of iterated

integrals over square root valued letters representing incomplete elliptic integrals and iterations over them.

The expansion in the limit $m_e^2 \ll s$ reveals errors in the constant $O(\alpha^2)$ term of the former calculation

and yields agreement with the calculation based on massive operator matrix elements, which has impact on the

experimental analysis programs. This finding also explicitly proofs the

factorization of massive initial state particles in the high energy limit at $\mathcal{O}(\alpha^2)$ for this process.