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Evaluation of Fermi-Dirac integral. (English) Zbl 0673.65010
Nonlinear numerical methods and rational approximation, Proc. Conf., Antwerp/Belgium 1987, Math. Appl., D. Reidel Publ. Co. 43, 435-444 (1988).

[For the entire collection see Zbl 0658.00009.]

The author considers general methods for the Fermi-Dirac integral ${F}_{\mu }\left(z\right)={\int }_{0}^{z}{x}^{\mu }{\left(1+exp\left(x-z\right)\right)}^{-1}dx,$ $\mu >-1$ and distinguishes three cases: (i) $\mu =1,2,···,z\ge 0$ and the non- polynomial part of ${F}_{\mu }\left(z\right)$ is expanded, after a suitable variable transformation, into Chebyshev series; $\left(ii\right)\phantom{\rule{1.em}{0ex}}\mu =-,,···,z\ge 0$ is sufficiently small and ${F}_{\mu }\left(z\right)$ is expanded in powers of $x=1-{\left(1+{e}^{z}\right)}^{1/2};$ (iii) $\mu =-,,···,z\ge {u}^{2},$ with a sufficiently large u and ${F}_{\mu }\left(z\right)$ is expanded at first into a series containing the functions Erfi and Erfc and, after that, into Chebyshev series with the variable u/$\sqrt{z}$.

Reviewer: R.S.Dahiya
##### MSC:
 65D20 Computation of special functions, construction of tables 33E99 Other special functions
##### Keywords:
Fermi-Dirac integral; Chebyshev series