The evaluation of Fermi-Dirac integrals is discussed for cases in which the Sommerfeld method fails. Such cases occur when the integrand has a singularity at the Fermi surface and when the integrand is a rapidly oscillating function. As examples, the first-order exchange integral for electrons and the free-energy integral of the noninteracting electron gas in a magnetic field are evaluated. The method uses a contour-integral representation of the Fermi function (previously mentioned by Dingle), supplemented by Mittag-Leffler type expansions.
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Wasserman et al. (1970) studied this question.
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