Complete Fermi–Dirac integral

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In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index is defined by

This equals

where is the polylogarithm.

Its derivative is

and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for appears in the literature, for instance some authors omit the factor . The definition used here matches that in the NIST DLMF.

Special values[edit]

The closed form of the function exists for j = 0:

See also[edit]


  • Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "3.411.3.". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. p. 355. ISBN 0-12-384933-0. LCCN 2014010276. ISBN 978-0-12-384933-5.
  • R.B.Dingle (1957). Fermi-Dirac Integrals. Appl.Sci.Res. B6. pp. 225–239.

External links[edit]