From Wikipedia, the free encyclopedia
In mathematics, the incomplete Fermi-Dirac integral, named after Enrico Fermi and Paul Dirac, for an index
and parameter
is given by
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a7d6bf1a96c51f7b62afa1d7f805d88d36f28cb)
Its derivative is
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {F} _{j}(x,b)=\operatorname {F} _{j-1}(x,b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb797de79629feb55e11710fc030d59a260c978)
and this derivative relationship may be used to find the value of the incomplete Fermi-Dirac integral for non-positive indices
.[1]
This is an alternate definition of the incomplete polylogarithm, since:
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6510761a044c04a6bc8df2ef57ef4a41902eb78)
Which can be used to prove the identity:
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ce9f6adb88f42dac128c2677f2122c37b34b95)
where
is the gamma function and
is the upper incomplete gamma function. Since
, it follows that:
![{\displaystyle \operatorname {F} _{j}(x,0)=\operatorname {F} _{j}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05eca4a7e9ab5b1315dae70b96118cba7127da35)
where
is the complete Fermi-Dirac integral.
The closed form of the function exists for
: [1]
![{\displaystyle \operatorname {F} _{0}(x,b)=\ln \!{\big (}1+e^{x-b}{\big )}-(b-x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3973bd279e7bd04da2c020e6fbcfa9505143e5)