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

${\mathrm{F}}_j(x,b)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{\mathrm{\Gamma }(j+1)}\underset{b}{\overset{\mathrm{\infty }}{\int }}\frac{t^j}{e^{t-x}+1}\mathrm{d}t$

Its derivative is

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

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:

${\mathrm{F}}_j(x,b)=\frac{1}{\mathrm{\Gamma }(j+1)}\underset{b}{\overset{\mathrm{\infty }}{\int }}\frac{t^j}{e^{t-x}+1}\mathrm{d}t=\frac{1}{\mathrm{\Gamma }(j+1)}\underset{b}{\overset{\mathrm{\infty }}{\int }}\frac{t^j}{\frac{e^t}{e^x}+1}\mathrm{d}t=-\frac{1}{\mathrm{\Gamma }(j+1)}\underset{b}{\overset{\mathrm{\infty }}{\int }}\frac{t^j}{\frac{e^t}{-e^x}-1}\mathrm{d}t=-{\mathrm{Li}}_{j+1}(b,-e^x)$

Which can be used to prove the identity:

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

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

${\mathrm{F}}_j(x,0)={\mathrm{F}}_j(x)$

where ${\mathrm{F}}_j(x)$ is the complete Fermi-Dirac integral.

Special values

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

${\mathrm{F}}_0(x,b)=\ln (1+e^{x-b})$

See also

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

External links

• GNU Scientific Library - Reference Manual

• Weisstein, Eric W.. "Fermi-Dirac distribution". http://mathworld.wolfram.com/Fermi-DiracDistribution.html.