In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j is defined by
F j ( x ) = 1 Γ ( j + 1 ) ∫ 0 ∞ t j e t − x + 1 d t , ( j > − 1 ) {\displaystyle F_{j}(x)={\frac {1}{\Gamma (j+1)}}\int _{0}^{\infty }{\frac {t^{j}}{e^{t-x}+1}}\,dt,\qquad (j>-1)}
This equals
− Li j + 1 ( − e x ) , {\displaystyle -\operatorname {Li} _{j+1}(-e^{x}),}
where Li s ( z ) {\displaystyle \operatorname {Li} _{s}(z)} is the polylogarithm. Its derivative is
d F j ( x ) d x = F j − 1 ( x ) , {\displaystyle {\frac {dF_{j}(x)}{dx}}=F_{j-1}(x),}
and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for F j {\displaystyle F_{j}} appears in the literature, for instance some authors omit the factor 1 / Γ ( j + 1 ) {\displaystyle 1/\Gamma (j+1)} . The definition used here matches that in the NIST DLMF.
Special values The closed form of the function exists for j = 0:
F 0 ( x ) = ln ( 1 + exp ( x ) ) . {\displaystyle F_{0}(x)=\ln(1+\exp(x)).}
For x = 0, the result reduces to
F j ( 0 ) = η ( j + 1 ) , {\displaystyle F_{j}(0)=\eta (j+1),}
where η {\displaystyle \eta } is the Dirichlet eta function.
See also Incomplete Fermi–Dirac integral Gamma function Polylogarithm
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.