Incomplete Fermi–Dirac integral

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

${\displaystyle \operatorname {F} _{j}(x,b){\overset {\mathrm {def} }{=}}{\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }\!{\frac {t^{j}}{e^{t-x}+1}}\;\mathrm {d} t}$

Its derivative is

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {F} _{j}(x,b)=\operatorname {F} _{j-1}(x,b)}$

and this derivative relationship may be used to find the value of the incomplete Fermi-Dirac integral for non-positive indices ${\displaystyle j}$.^[1]

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

${\displaystyle \operatorname {F} _{j}(x,b)={\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }\!{\frac {t^{j}}{e^{t-x}+1}}\;\mathrm {d} t={\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }\!{\frac {t^{j}}{\displaystyle {\frac {e^{t}}{e^{x}}}+1}}\;\mathrm {d} t=-{\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }\!{\frac {t^{j}}{\displaystyle {\frac {e^{t}}{-e^{x}}}-1}}\;\mathrm {d} t=-\operatorname {Li} _{j+1}(b,-e^{x})}$

Which can be used to prove the identity:

${\displaystyle \operatorname {F} _{j}(x,b)=-\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{j+1}}}{\frac {\Gamma (j+1,nb)}{\Gamma (j+1)}}e^{nx}}$

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

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

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

Special values

The closed form of the function exists for ${\displaystyle j=0}$: ^[1]

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

See also

• Complete Fermi–Dirac integral
• Fermi–Dirac statistics
• Incomplete polylogarithm
• Polylogarithm