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Analytical expansion and numerical approximation of the Fermi-Dirac integrals \(\mathcal F_j(x)\) of order \(j=-1/2\) and \(j=1/2\). (English) Zbl 0987.65024
This paper uses properties of the Weyl semiintegral and semiderivative, along with Oldham’s representations of the Randle-Sevcik function from electrochemistry, to derive finite series expansions for the Fermi-Dirac integrals \({\mathcal F}_j(x)\), \(-\infty< x< \infty\), \(j= 1/2,1/2\). The practical use of these expansions for the numerical approximation of \({\mathcal F}_{-1/2}(x)\) and \({\mathcal F}_{1/2}(x)\) over the finite intervals is investigated and an extension of these results to the higher-order cases \(j= 3/2\), \(5/2\), \(7/2\) is outlined.

MSC:
65D20 Computation of special functions and constants, construction of tables
33E20 Other functions defined by series and integrals
Software:
Fermi-Dirac
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