# Fermi–Dirac integrals

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## 1—10 of 17 matching pages

##### 1: 25.19 Tables
• Morris (1979) tabulates $\operatorname{Li}_{2}\left(x\right)$25.12(i)) for $\pm x=0.02(.02)1(.1)6$ to 30D.

• Cloutman (1989) tabulates $\Gamma\left(s+1\right)F_{s}(x)$, where $F_{s}(x)$ is the FermiDirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

• ##### 2: 25.20 Approximations
• Morris (1979) gives rational approximations for $\operatorname{Li}_{2}\left(x\right)$25.12(i)) for $0.5\leq x\leq 1$. Precision is varied with a maximum of 24S.

• Antia (1993) gives minimax rational approximations for $\Gamma\left(s+1\right)F_{s}(x)$, where $F_{s}(x)$ is the FermiDirac integral (25.12.14), for the intervals $-\infty and $2\leq x<\infty$, with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$. For each $s$ there are three sets of approximations, with relative maximum errors $10^{-4},10^{-8},10^{-12}$.

• ##### 3: 25.12 Polylogarithms
The FermiDirac and Bose–Einstein integrals are defined by
25.12.14 $F_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{e^{t-x}% +1}\,\mathrm{d}t,$ $s>-1$,
In terms of polylogarithms
$F_{s}(x)=-\operatorname{Li}_{s+1}\left(-e^{x}\right),$
For a uniform asymptotic approximation for $F_{s}(x)$ see Temme and Olde Daalhuis (1990).
##### 4: 25.18 Methods of Computation
For dilogarithms and polylogarithms see Jacobs and Lambert (1972), Osácar et al. (1995), Spanier and Oldham (1987, pp. 231–232), and Zudilin (2007). For FermiDirac and Bose–Einstein integrals see Cloutman (1989), Gautschi (1993), Mohankumar and Natarajan (1997), Natarajan and Mohankumar (1993), Paszkowski (1988, 1991), Pichon (1989), and Sagar (1991a, b). …
##### 5: Bibliography P
• S. Paszkowski (1988) Evaluation of Fermi-Dirac Integral. In Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987), A. Cuyt (Ed.), Mathematics and Its Applications, Vol. 43, pp. 435–444.
• S. Paszkowski (1991) Evaluation of the Fermi-Dirac integral of half-integer order. Zastos. Mat. 21 (2), pp. 289–301.
• B. Pichon (1989) Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.
• ##### 6: Bibliography T
• N. M. Temme and A. B. Olde Daalhuis (1990) Uniform asymptotic approximation of Fermi-Dirac integrals. J. Comput. Appl. Math. 31 (3), pp. 383–387.
• J. S. Thompson (1996) High Speed Numerical Integration of Fermi Dirac Integrals. Master’s Thesis, Naval Postgraduate School, Monterey, CA.
• A. Trellakis, A. T. Galick, and U. Ravaioli (1997) Rational Chebyshev approximation for the Fermi-Dirac integral $F_{-3/2}(x)$ . Solid–State Electronics 41 (5), pp. 771–773.
##### 8: Bibliography N
• A. Natarajan and N. Mohankumar (1993) On the numerical evaluation of the generalised Fermi-Dirac integrals. Comput. Phys. Comm. 76 (1), pp. 48–50.
• ##### 9: Bibliography F
• L. W. Fullerton and G. A. Rinker (1986) Generalized Fermi-Dirac integrals—FD, FDG, FDH. Comput. Phys. Comm. 39 (2), pp. 181–185.
• ##### 10: Bibliography S
• R. P. Sagar (1991a) A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals. Comput. Phys. Comm. 66 (2-3), pp. 271–275.
• R. P. Sagar (1991b) On the evaluation of the Fermi-Dirac integrals. Astrophys. J. 376 (1, part 1), pp. 364–366.