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

This equals

where is the polylogarithm.

Its derivative is

and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for appears in the literature, for instance some authors omit the factor . The definition used here matches that in the NIST DLMF.

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

For x = 0, the result reduces to

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