Английская Википедия:Complete Fermi–Dirac integral

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

Шаблон:Use American English Шаблон:Short description In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j  is defined by

<math>F_j(x) = \frac{1}{\Gamma(j+1)} \int_0^\infty \frac{t^j}{e^{t-x} + 1}\,dt, \qquad (j > -1)</math>

This equals

<math>-\operatorname{Li}_{j+1}(-e^x),</math>

where <math>\operatorname{Li}_{s}(z)</math> is the polylogarithm.

Its derivative is

<math>\frac{dF_{j}(x)}{dx} = F_{j-1}(x) , </math>

and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for <math>F_j</math> appears in the literature, for instance some authors omit the factor <math>1/\Gamma(j+1)</math>. The definition used here matches that in the NIST DLMF.

Special values

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

<math>F_0(x) = \ln(1+\exp(x)).</math>

For x = 0, the result reduces to

<math> F_j(0) = \eta(j+1), </math>

where <math>\eta</math> is the Dirichlet eta function.

See also

References

External links


Шаблон:Mathanalysis-stub

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Complete_Fermi–Dirac_integral&oldid=14038944