Complete Fermi–Dirac integral explained

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
	\Gamma(j+1)

	t^j
	e^t-x+1

dt, (j>-1)

This equals

-\operatorname{Li}_j+1(-e^x),

where

\operatorname{Li}_s(z)

Its derivative is

	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

appears in the literature, for instance some authors omit the factor

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)).

For x = 0, the result reduces to

F_j(0)=η(j+1),

where

Book: Izrail Solomonovich . Gradshteyn . Izrail Solomonovich Gradshteyn . Iosif Moiseevich . Ryzhik . Iosif Moiseevich Ryzhik . Yuri Veniaminovich . Geronimus . Yuri Veniaminovich Geronimus . Michail Yulyevich . Tseytlin . Michail Yulyevich Tseytlin . Alan . Jeffrey . Daniel . Zwillinger . Victor Hugo . Moll . Victor Hugo Moll . Scripta Technica, Inc. . Table of Integrals, Series, and Products . . 2015 . October 2014 . 8 . English . 978-0-12-384933-5 . . 2014010276 . 2016-02-21-->. Gradshteyn and Ryzhik . 3.411.3. . 355.
Book: Fermi-Dirac Integrals. R.B.Dingle. Appl.Sci.Res. B6. 1957. 225–239.