Incomplete Fermi–Dirac integral
In mathematics, the incomplete Fermi-Dirac integral, named after Enrico Fermi and Paul Dirac, for an index [math]\displaystyle{ j }[/math] and parameter [math]\displaystyle{ b }[/math] is given by
- [math]\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 }[/math]
Its derivative is
- [math]\displaystyle{ \frac{\mathrm{d}}{\mathrm{d}x}\operatorname{F}_j(x,b) = \operatorname{F}_{j-1}(x,b) }[/math]
and this derivative relationship is used to define the incomplete Fermi-Dirac integral for non-positive indices [math]\displaystyle{ j }[/math].
This is an alternate definition of the incomplete polylogarithm, since:
- [math]\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) }[/math]
Which can be used to prove the identity:
- [math]\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} }[/math]
where [math]\displaystyle{ \Gamma(s) }[/math] is the gamma function and [math]\displaystyle{ \Gamma(s,y) }[/math] is the upper incomplete gamma function. Since [math]\displaystyle{ \Gamma(s,0)=\Gamma(s) }[/math], it follows that:
- [math]\displaystyle{ \operatorname{F}_j(x,0) = \operatorname{F}_j(x) }[/math]
where [math]\displaystyle{ \operatorname{F}_j(x) }[/math] is the complete Fermi-Dirac integral.
Special values
The closed form of the function exists for [math]\displaystyle{ j=0 }[/math]:
- [math]\displaystyle{ \operatorname{F}_0(x,b) = \ln\!\big(1+e^{x-b}\big) }[/math]
See also
External links
- Weisstein, Eric W.. "Fermi-Dirac distribution". http://mathworld.wolfram.com/Fermi-DiracDistribution.html.
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