Computing Dirac’s atomic hydrogen wave functions of the continuum, using summation of mathematically divergent series

Accurate computation of the positive and negative energy wave functions for the continuous parts of the atomic hydrogen spectrum, at large distances from the proton, provides an example of the problems in which sums of mathematically divergent series have to be computed. Here it is shown how such summations can be performed and what checks on the accuracy of the methods are used. Similar methods should be applicable in many other cases.