Mathematical integral
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
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