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.
"Eichler integrals for Maass cusp forms of half-integral weight." Illinois J. Math. 57 (2) 445 - 475, Summer 2013. https://doi.org/10.1215/ijm/1408453590