Weak Maass-Poincare 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 Borcherd-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. For the purpose, we use a weak Maass-Poincar\'e Series. The secondary goal is to provide a full computation of the Fourier coefficients for the weak Maass-Poincar\'e Series in most general form as a weak Maass-Poincar\'e Series has played a key role in the recent advances in the theory of weak Maass forms.