A calculational procedure of the Fermi–Dirac integral with an arbitrary real index by means of a numerical integration technique

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Abstract
A calculational procedure of the Fermi–Dirac integral F(r,η)=∫∞0xr/(ex−η+1)dx with an arbitrary real index r by means of a numerical integration technique is proposed. Numerical values of F(r,η) can be calculated by applying an asymptotic series expansion to the integrand xr/(ex−η+1) for the domain 0≤x≤0.1 and by approximating the infinite upper limit at L=a+br+η, where a and b are constants estimated from the allowable calculational error. The total calculational error can be reduced to 10−7 at any η for − (1)/(2) ≤r≤10 by employing the constants a=17 and b=4.5 and using the first four terms of the series expansion.

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