Complete Fermi–Dirac integral

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

$F_j(x) = \frac{1}{\Gamma(j+1)} \int_0^\infty \frac{t^j}{\exp(t-x) + 1}\,dt.$

This is an alternate definition of the polylogarithm function. The closed form of the function exists for j = 0:

$F_0(x) = \ln(1+\exp(x)).\,$

See also

• Incomplete Fermi–Dirac integral
• Gamma function
• Table of Integrals, Series, and Products, I.S. Gradshteyn, I.M. Ryzhik, 5th edition, p. 370, formula № 3.411.3.

External links

• GNU Scientific Library - Reference Manual
• Fermi-Dirac integral calculator for iPhone/iPad

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