Complete Fermi–Dirac Integral

In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index ''j '' is defined by

:$F_j(x) = \frac \int_0^\infty \frac\,dt, \qquad (j > -1)$

This equals
:$-\operatorname_(-e^x),$
where $\operatorname_(z)$ is the polylogarithm.

Its derivative is
:$\frac = F_(x) ,$
and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices ''j''. Differing notation for $F_j$ appears in the literature, for instance some authors omit the factor $1/\Gamma(j+1)$. The definition used here matches that in th
NIST DLMF

Special values

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

:$F_0(x) = \ln(1+\exp(x)).$

For ''x = 0'', the result reduces to

$F_j(0) = \eta(j+1),$

where $\eta$ is the Dirichlet eta function.

See also

* Incomplete Fermi–Dirac integral
* Gamma function
* Polylogarithm

References

*
*

External links

GNU Scientific Library - Reference Manual

Fermi-Dirac integral calculator for iPhone/iPad

Notes on Fermi-Dirac Integrals

Section in NIST Digital Library of Mathematical Functions

npplus
Python package that provides (among others) Fermi-Dirac integrals and inverses for several common orders.


Definition given by Wolfram's MathWorld.

Special functions

{{mathanalysis-stub