Complete Fermi–Dirac integral - Alchetron, the free social encyclopedia

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 > 0 )

This equals

− Li j + 1 ⁡ ( − e x ) ,

where Li s ⁡ ( z ) is the polylogarithm.

Its derivative is

d F j ( x ) d x = F j − 1 ( x ) ,

and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j.

Special values

The closed form of the function exists for j = 0:

F 0 ( x ) = ln ⁡ ( 1 + exp ⁡ ( x ) ) .

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