Weak Maass–Poincaré series and weight 3/2 mock modular forms

The primary goal of this paper is to construct the basis of the space of weight 3/2 mock modular forms, which is an extension of the Borcherds–Zagier basis of weight 3/2 weakly holomorphic modular forms. The shadows of the members of this basis form the Borcherds–Zagier basis of the space of weight 1/2 weakly holomorphic modular forms. In the course of the construction, we provide a full computation of the Fourier coefficients for the weak Maass–Poincaré series in most general form for the purpose of future reference.