  The cross section for different processes induced by $e^+e^-$ annihilation,
in the kinematical limit
$\beta_{\mu}\approx\beta_{\pi}=(1-m_{\pi}^2/\epsilon^2)^{1/2}\sim 1$, is
calculated taking into account first order corrections to the amplitudes and
the corrections due to soft emitted photons, with energy $\omega\le\Delta E\le
\epsilon$ in the center of mass of the $e^+e^-$ colliding beams. The results
are given separately for charge--odd and charge--even terms in the final
channels $\pi^+\pi^-(\gamma)$ and $\mu^+\mu^-(\gamma)$. In case of pions, form
factors are taken into account. The differential cross sections for the
processes: $e^++e^-\to e^++e^-(+\gamma)$, $\to \pi^++\pi^-(\gamma)$, $\to
\mu^++\mu^-(\gamma),\to \gamma\gamma(\gamma)$ have been calculated and the
corresponding formula are given in the ultrarelativistic limit $\sqrt{s}/2=
\epsilon \gg m_{\mu}\sim m_{\pi}$ . For a quantitative evaluation of the
contribution of higher order of the perturbation theory, the production of
$\pi^+\pi^-$, including radiative corrections, is calculated in the approach of
the lepton structure functions. This allows to estimate the precision of the
obtained results as better than 0.5% outside the energy region corresponding to
narrow resonances. A method to integrate the cross section, avoiding the
difficulties which arise from singularities is also described.
