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 is used to define 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 ) {\displaystyle \operatorname {F} _{0}(x,b)=\ln \!{\big (}1+e^{x-b}{\big )}}

See also

External links

  • GNU Scientific Library - Reference Manual
  • Weisstein, Eric W. "Fermi-Dirac distribution". MathWorld.


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