Incomplete Fermi–Dirac integral

In mathematics, the incomplete Fermi-Dirac integral, named after Enrico Fermi and Paul Dirac, for an index $j$ and parameter $b$ is given by

\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

{\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 $j$ .

This is an alternate definition of the incomplete polylogarithm, since:

\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:

\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 $\Gamma (s)$ is the gamma function and $\Gamma (s,y)$ is the upper incomplete gamma function. Since $\Gamma (s,0)=\Gamma (s)$ , it follows that:

\operatorname {F} _{j}(x,0)=\operatorname {F} _{j}(x)

where $\operatorname {F} _{j}(x)$ is the complete Fermi-Dirac integral.

Special values edit

The closed form of the function exists for $j=0$ :

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

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