Has the Fermi-Dirac Integral been solved?

[patric44]

TL;DR Summary

has the Fermi-Dirac Integral been solved?

hi guys
I have a question about whether or not the Fermi-Dirac Integral has Been solved, because i found a formula on Wikipedia that relates the Fermi-Dirac integral with the polylogarithm function:
$$F_{j}(x) = -Li_{j+1}(-e^{x})$$
and in some recent papers they claim that no analytical solution exist, plus if the formula on Wikipedia is correct why there are some recent papers discussing a Numerical solution for particular cases of the Fermi-Dirac integral?

 

[jasonRF]

The formula is probably correct since it agrees with DLMF
https://dlmf.nist.gov/25.12#iii
which is very reliable. However, computing the polylogarithm is not easy in general, and I’m not surprised that you can find papers discussing different numerical methods. The only time I have needed polylogarithms I used the integral definition
https://dlmf.nist.gov/25.12#E11
along with generalized Gauss-Laguerre integration.
https://en.m.wikipedia.org/wiki/Gauss-Laguerre_quadrature
It worked fine for the parameter range I cared about, but it is probably not a general approach that is practical for all situations.

Jason

 

[dextercioby]

Well, writing the F-D integral (or the B-E integral) in terms of known special functions (most of them essentially reducible to hyper-geometric functions or Meijer functions) is an analytical solution, if the series expansion is convergent. You can't expect that these integrals are expressible in a finite combination of elementary functions...