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A Model for Fermi-Dirac Integrals

​
k
/2
-1
1
3
5
Fermi–Dirac integrals arise in calculating pressure and density in degenerate matter, such as neutron stars; they also occur in the electronic density of semiconductors. A (semi-)closed form was not known until 1995, when Howard Lee noticed the application of the integral form of polylogarithms. We developed an alternative expression,
F
k/2
(η)=
1
Γ1+
k
2
∞
∫
0
k/2
x
1+
x-η
e
dx
(plotted here), using an exponential model that is accurate for
η<3
(arXiv:0909.3653v5 [math.GM]). Here we compare the model, in green, with the polylogarithm expression, in blue, for Fermi–Dirac integrals half-integer order
k
2
, with
k=-1,1,3,5
(orders commonly used in astrophysics and semiconductors).
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