Modified Sommerfeld-Maue wave function: Application to the pair-production process

A solution of the Dirac equation for an electron in a point-Coulomb potential is given in the form of an expansion in the parameter $\frac{1}{r}$. The main part of the solution is the Sommerfeld-Maue wave function. We introduce a correction term. All orders in $\alpha Z$ are kept. The given solution (Sommerfeld-Maue plus correction term) contains the exact contributions of orders 0 and 1 in the expansion in $\frac{1}{r}$. This modified Sommerfeld-Maue wave function is used to get the cross section for the pair-production process by $\gamma$ rays in the field of a nucleus. The cross section is written in the form of the Fink-Pratt cross section plus a correction term. The formula is valid for all $\alpha Z$. The energetic relative accuracy should be of order $\frac{1}{E^2}$. The small-angle approximation is not introduced.