expression for $F(\omega, \zeta)$ in (\[3.14\]) coincides with the function $F(\omega, \zeta)$ defined in [@BKS1], Eq.(13). The second (subtraction) term in (\[3.14\])which is naturally arisen in this derivation was missed in Eq.(13), [@BKS1] as it was said above. The expression (\[3.11\]) is consistent with Eq.(21.6) in the book [@BKS3] (see also Eq.(2.2) in [@BKS2]) where another physical problem was analyzed. It is the incoherent bremsstrahlung in the oriented crystals.

Below we restrict ourselves to the case of unpolarized electrons and photons. Influence of bounded transverse size on the probability of process with polarized particles will be considered elsewhere.

Gaussian beams
==============

For calculation of explicit expression for the bremsstrahlung cross section we have to specify the distributions of particles in the colliding beams. Here we consider the actual case of Gaussian beams. Using the Fourier transform we have $$\begin{aligned}
\hspace{-10mm}&& w({\bf x})=\frac{1}{(2\pi)^2}\int d^2q
\exp(-i{\bf q x})w(\bf q);
\nonumber \\
\hspace{-10mm}&& w_r({\bf
q})=\exp\left[-\frac{1}{2}(q_z^2\Delta_z^2+q_y^2\Delta_y^2)\right],~
w_c({\bf
q})=\exp\left[-\frac{1}{2}(q_z^2\sigma_z^2+q_y^2\sigma_y^2)\right],
\label{4.1}\end{aligned}$$ where as above the index $r$ relates to the radiating beam and the index $c$ relates to the target beam, $\Delta_z$ and $\Delta_y$ ($\sigma_z$ and $\sigma_y$) are the vertical and horizontal transverse dimensions of radiating (target) beam. Substituting (\[4.1\]) into Eq.(\[3.15\]) we find $$\begin{aligned}
\hspace{-5mm}&& 