Radial integrals for electron pair production in a point-Coulomb potential

A theoretical formalism is given for the calculation of electron pair production in a point-Coulomb potential using the method developed by Sud, Wright, and Onley for expressing the radial integrals over the product of Dirac-Coulomb functions in terms of a matrix generalization of the $\Gamma$ function. A partial differential equation in lepton energy satisfied by the $\Gamma$ function is obtained. This partial differential equation can be integrated numerically to obtain the positron spectrum for high-energy photons.