On the evaluation of Fermi-Dirac integral and its derivatives by IMT and DE quadrature methods

The numerical values of the derivatives of the Fermi-Dirac integral, up to the third order, with respect to its parameters are evaluated by both the IMT and the Double Exponential integration methods. These integrals find extensive use in astrophysical problems. It is found that both the IMT and the DE methods are very efficient, especially for handling the end point integrable singularities. Also, the DE method is slightly superior to the IMT method. Unlike the IMT or Gauss methods, the nodes of DE method can be evaluated in terms of built in elementary functions, which is a big advantage.