Incomplete Fermi–Dirac integral explained

In mathematics, the incomplete Fermi-Dirac integral, named after Enrico Fermi and Paul Dirac, for an index

j and parameter b is given by\operatorname{F} _{j(x,b)} \overset{ def } \frac \int_b^\infty\! \frac\;\mathrmt

Its derivative is

(d)/(d x) \operatorname{F} _{j(x,b)} = \operatorname{F} _{j-1} (x,b) and this derivative relationship may be used to find the value of the incomplete Fermi-Dirac integral for non-positive indices j .^[1]

This is an alternate definition of the incomplete polylogarithm, since:

\operatorname{F} _{j(x,b)} = (1)/(\Gamma(j +1 )) \int ^{inf ty} _{b} (t ^{j})/(e ^{t-x} +1) d t = (1)/(\Gamma(j +1 )) \int ^{inf ty} _{b} (t ^{j})/(\displaystyle (e ^{t})/(e ^{x}) + 1) d t =- (1)/(\Gamma(j +1 )) \int ^{inf ty} _{b} (t ^{j})/(\displaystyle (e ^{t})/(- e ^{x}) - 1) d t = - \operatorname{Li} _{j+1} (b, - e ^{x)}

Which can be used to prove the identity:

\operatorname{F} _{j(x,b)} = - \sum ^{inf ty} _{n =1} (( -1 ) ^{n})/(n ^{j+1}) (\Gamma(j +1 ,nb))/(\Gamma(j +1 )) e ^{nx} where \Gamma(s) is the gamma function and \Gamma(s,y) is the upper incomplete gamma function. Since \Gamma(s, 0 ) = \Gamma(s) , it follows that:\operatorname{F} _{j(x,0)} = \operatorname{F} _{j(x)}

where

\operatorname{F} _{j(x)} is the complete Fermi-Dirac integral.

Special values

The closed form of the function exists for

j =0 : ^[1] \operatorname{F} _{0(x,b)} = ln ( 1+ e ^{x-b} ) - (b - x)

See also

• Complete Fermi–Dirac integral
• Fermi–Dirac statistics
• Incomplete polylogarithm
• Polylogarithm

External links

• GNU Scientific Library - Reference Manual

Notes and References

1. Guano . Michele . Algorithm 745: computation of the complete and incomplete Fermi-Dirac integral. . ACM Transactions on Mathematical Software . 1995 . 21 . 3 . 221–232 . 10.1145/210089.210090 . 26 June 2024.