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 may be used to find the value of 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
The closed form of the function exists for $$j=0$$:


 * $$\operatorname{F}_0(x,b) = \ln\!\big(1+e^{x-b}\big) - (b - x) $$