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

:<math>F_j(x) = \frac{1}{\Gamma(j+1)} \int_0^\infty \frac{t^j}{e^{t-x} + 1}\,dt, \qquad (j > -1)</math>

This equals

:<math>-\operatorname{Li}_{j+1}(-e^x),</math>

where <math>\operatorname{Li}_{s}(z)</math> is the polylogarithm.

Its derivative is

:<math>\frac{dF_{j}(x)}{dx} = F_{j-1}(x) , </math>

and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for <math>F_j</math> appears in the literature, for instance some authors omit the factor <math>1/\Gamma(j+1)</math>. The definition used here matches that in the NIST DLMF.

Special values

The closed form of the function exists for j&nbsp;=&nbsp;0:

:<math>F_0(x) = \ln(1+\exp(x)).</math>

For x = 0, the result reduces to

<math> F_j(0) = \eta(j+1), </math>

where <math>\eta</math> is the Dirichlet eta function.

See also

  • Incomplete Fermi–Dirac integral
  • Gamma function
  • Polylogarithm

References

  • 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.
  • Wolfram's MathWorld: Definition given by Wolfram's MathWorld.