Incomplete Fermi–Dirac integral

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

Contents

Its derivative is

and this derivative relationship may be used to find the value of the incomplete Fermi-Dirac integral for non-positive indices . [1]

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

Which can be used to prove the identity:

where is the gamma function and is the upper incomplete gamma function. Since , it follows that:

where is the complete Fermi-Dirac integral.

Special values

The closed form of the function exists for : [1]

See also


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References

  1. 1 2 Guano, Michele (1995). "Algorithm 745: computation of the complete and incomplete Fermi-Dirac integral". ACM Transactions on Mathematical Software. 21 (3): 221–232. doi:10.1145/210089.210090 . Retrieved 26 June 2024.