Complete Fermi–Dirac integral
In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j 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
The closed form of the function exists for j = 0:
For x = 0, the result reduces to
where is the Dirichlet eta function.
References
- 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 978-0-12-384933-5. LCCN 2014010276. ISBN 978-0-12-384933-5.
- R.B.Dingle (1957). Fermi-Dirac Integrals. Appl.Sci.Res. B6. pp. 225–239.
External links
- GNU Scientific Library - Reference Manual
- Fermi-Dirac integral calculator for iPhone/iPad
- Notes on Fermi-Dirac Integrals
- Section in NIST Digital Library of Mathematical Functions
- npplus: Python package that provides (among others) Fermi-Dirac integrals and inverses for several common orders.
- Wolfram's MathWorld: Definition given by Wolfram's MathWorld.
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