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 }\!{\frac {t^{j}}{e^{t-x}+1}}\;\mathrm {d} t={\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }\!{\frac {t^{j}}{\displaystyle {\frac {e^{t}}{e^{x}}}+1}}\;\mathrm {d} t=-{\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }\!{\frac {t^{j}}{\displaystyle {\frac {e^{t}}{-e^{x}}}-1}}\;\mathrm {d} t=-\operatorname {Li} _{j+1}(b,-e^{x})}
Which can be used to prove the identity:
F j ( x , b ) = − ∑ n = 1 ∞ ( − 1 ) n n j + 1 Γ ( j + 1 , n b ) Γ ( j + 1 ) e n x {\displaystyle \operatorname {F} _{j}(x,b)=-\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{j+1}}}{\frac {\Gamma (j+1,nb)}{\Gamma (j+1)}}e^{nx}}
where Γ ( s ) {\displaystyle \Gamma (s)} is the gamma function and Γ ( s , y ) {\displaystyle \Gamma (s,y)} is the upper incomplete gamma function. Since Γ ( s , 0 ) = Γ ( s ) {\displaystyle \Gamma (s,0)=\Gamma (s)} , it follows that:
F j ( x , 0 ) = F j ( x ) {\displaystyle \operatorname {F} _{j}(x,0)=\operatorname {F} _{j}(x)}
where F j ( x ) {\displaystyle \operatorname {F} _{j}(x)} is the complete Fermi-Dirac integral.
Special values The closed form of the function exists for j = 0 {\displaystyle j=0} :
F 0 ( x , b ) = ln ( 1 + e x − b ) − ( b − x ) {\displaystyle \operatorname {F} _{0}(x,b)=\ln \!{\big (}1+e^{x-b}{\big )}-(b-x)}
See also Complete Fermi–Dirac integral Fermi–Dirac statistics Incomplete polylogarithm Polylogarithm
References
External links GNU Scientific Library - Reference Manual Weisstein, Eric W. "Fermi-Dirac distribution". MathWorld.