Incomplete Fermi–Dirac integral
Template:Onesource In mathematics, the incomplete Fermi-Dirac integral, named after Enrico Fermi and Paul Dirac, for an index <math>j</math> and parameter <math>b</math> is given by
- <math>\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</math>
Its derivative is
- <math>\frac{\mathrm{d}}{\mathrm{d}x}\operatorname{F}_j(x,b) = \operatorname{F}_{j-1}(x,b) </math>
and this derivative relationship may be used to find the value of the incomplete Fermi-Dirac integral for non-positive indices <math>j</math>.<ref name = "Guano, M.">Template:Cite journal</ref>
This is an alternate definition of the incomplete polylogarithm, since:
- <math>\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) </math>
Which can be used to prove the identity:
- <math>
\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} </math> where <math>\Gamma(s)</math> is the gamma function and <math>\Gamma(s,y)</math> is the upper incomplete gamma function. Since <math>\Gamma(s,0)=\Gamma(s)</math>, it follows that:
- <math>\operatorname{F}_j(x,0) = \operatorname{F}_j(x)</math>
where <math>\operatorname{F}_j(x)</math> is the complete Fermi-Dirac integral.
Special valuesEdit
The closed form of the function exists for <math>j=0</math>: <ref name = "Guano, M."/>
- <math>\operatorname{F}_0(x,b) = \ln\!\big(1+e^{x-b}\big) - (b - x) </math>
See alsoEdit