Abstract: The plasma dispersion function (PDF) is widely used in plasma physics, particularly for the interaction between waves and plasmas. It is essentially important to quickly and precisely calculate the values of the PDF and its differentials with any orders. Related efforts have been ongoing since this function was first proposed until recent decades; many brilliant ideas have been developed, and some solutions behave well in specific situations. The paper aims to review the different calculation methods of the PDF and its differentials, and especially to compare their properties in numerical calculations, including the calculation accuracy, calculation speed and storage requirements. Then, some appropriate optimization numerical calculation methods can be recommended for different application scenarios.