完全费米—狄拉克积分

此處 $\operatorname {Li} _{s}(z)$ 為多重对数函数。

Fermi-Dirac Integral animation

Fermi-Dirac Integral complex minus

Fermi-Dirac Integral complex

完全费米—狄拉克积分，以恩里科·费米和保罗·狄拉克各取一字命名，已知指數j定义如下

F_{j}(x)={\frac {1}{\Gamma (j+1)}}\int _{0}^{\infty }{\frac {t^{j}}{e^{t-x}+1}}\,dt.

等於

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

特徵值

對j = 0，函數的封閉形式存在：

F_{0}(x)=\ln(1+\exp(x)).\,

當 $s=1$ ，與多重對數函數的值比較：

\operatorname {Li} _{1}(z)=-\log(1-z).

Table of Integrals, Series, and Products, I.S. Gradshteyn, I.M. Ryzhik, 5th edition, p. 370, formula № 3.411.3.

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