Complete Fermi–Dirac 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.

Source: Wikipedia — Complete Fermi–Dirac integral (CC BY-SA 4.0)

Complete Fermi–Dirac 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.

Source: Wikipedia "Complete Fermi–Dirac integral" · CC BY-SA 4.0

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