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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
Fermi-Dirac
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