  In this paper, we define and discuss Eichler integrals for Maass cusp forms
of half-integral weight on the full modular group. We discuss nearly periodic
functions associated to the Eichler integrals, introduce period functions for
such Maass cusp forms, and show that the nearly periodic functions and the
period functions are closely related. Those functions are extensions of the
periodic functions and period functions for Maass cusp forms of weight 0 on the
full modular group introduced by Lewis and Zagier.
