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Complete Fermi–Dirac integral

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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

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

External links


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