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Mathematical 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
[
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]
The closed form of the function exists for
j
= 0:
For
x = 0
, the result reduces to
where
is the
Dirichlet eta function
.
See also
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]
References
[
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
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
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]