View source for Incomplete Fermi–Dirac integral

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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.">{{cite journal |last1=Guano |first1=Michele |title=Algorithm 745: computation of the complete and incomplete Fermi-Dirac integral. |journal=ACM Transactions on Mathematical Software |date=1995 |volume=21 |issue=3 |pages=221–232 |doi=10.1145/210089.210090 |access-date=26 June 2024|url=https://dl.acm.org/doi/10.1145/210089.210090}}</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 values ==
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 also ==
* [[Complete Fermi–Dirac integral]]
* [[Fermi–Dirac statistics]]
* [[Incomplete polylogarithm]]
* [[Polylogarithm]]


==References==
{{reflist}}

==External links==
* [https://www.gnu.org/software/gsl/manual/gsl-ref.html#SEC119 GNU Scientific Library - Reference Manual]

* {{mathworld | urlname= Fermi-DiracDistribution | title= Fermi-Dirac distribution}}


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[[Category:Special functions]]


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