  The regular solutions to the Sch\"ordinger equation in the case of an
electron experiencing a Coulomb force are well known. Being that the radial
part of the differential equation to be solved is second order in derivatives,
it will have two independent solutions, with the second irregular solutions
being ill-behaved at the origin and unbound at infinity. For this reason,
second solutions are dropped for bound electrons. However, these second
solutions are still of academic interest for several reasons. One reason is to
help devise schemes to control numerical contamination by second solutions to
more general second-order differential equations when attempting to calculate
the first solutions through recursion relations. Another is to study the
analytic behavior of the electron Coulomb wave functions as the electron energy
and angular momentum are each extended into the complex plane, important in
investigations of bound-state poles and of Regge trajectories. In addition, toy
problems having a finite radial region with a Coulombic interaction, with
interior and exterior regions non Coulombic, require both regular and irregular
solutions to match solutions across the boundaries. In this presentation, exact
and closed form second irregular solutions are derived for the Coulomb bound
states, using a hither-to unnoticed trick in the Nikiforov-Uvorov method.
