Complete Fermi–Dirac integral

Short description: Mathematical integral

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

$F_j(x)=\frac{1}{\mathrm{\Gamma }(j+1)}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{t^j}{e^{t-x}+1}dt,(j>-1)$

This equals

$-{\mathrm{Li}}_{j+1}(-e^x),$

where ${\mathrm{Li}}_s(z)$ is the polylogarithm.

Its derivative is

$\frac{d{F}_j(x)}{dx}=F_{j-1}(x),$

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 $1/\mathrm{\Gamma }(j+1)$. The definition used here matches that in the 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

• "3.411.3." (in English). Table of Integrals, Series, and Products (8 ed.). Academic Press, Inc.. 2015. p. 355. ISBN:978-0-12-384933-5. ISBN 978-0-12-384933-5. 
• R.B.Dingle (1957). Fermi-Dirac Integrals. Appl.Sci.Res. B6. pp. 225–239. 

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.