Incomplete Fermi–Dirac integral

In mathematics, the incomplete Fermi-Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j {\displaystyle j} and parameter b {\displaystyle b} is given by F j ⁡ ( x , b ) = d e f 1 Γ ( j + 1 ) ∫ b ∞ t j e t − x + 1 d t {\displaystyle \operatorname {F} _{j}(x,b){\overset {\mathrm {def} }{=}}{\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }\! {\frac {t^{j}}{e^{t-x}+1}}\;\mathrm {d} t} Its derivative is d d x F j ⁡ ( x , b ) = F j − 1 ⁡ ( x , b ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {F} _{j}(x,b)=\operatorname {F} _{j-1}(x,b)} and this derivative relationship may be used to find the value of the incomplete Fermi-Dirac integral for non-positive indices j {\displaystyle j} . This is an alternate definition of the incomplete polylogarithm, since: F j ⁡ ( x , b ) = 1 Γ ( j + 1 ) ∫ b ∞ t j e t − x + 1 d t = 1 Γ ( j + 1 ) ∫ b ∞ t j e t e x + 1 d t = − 1 Γ ( j + 1 ) ∫ b ∞ t j e t − e x − 1 d t = − Li j + 1 ⁡ ( b , − e x ) {\displaystyle \operatorname {F} _{j}(x,b)={\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }\!

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