Fermi%E2%80%93Dirac%20integrals

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1: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►where the last integral denotes the Cauchy principal value (1.4.25). … ►If

f(t)

is absolutely integrable on

[0,R]

for every finite

R

, and the integral (1.14.47) converges, then … ►

§1.14(viii) Compendia

►For more extensive tables of the integral transforms of this section and tables of other integral transforms, see Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

2: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

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Other Integral Representations

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§8.19(ii) Graphics

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§8.19(x) Integrals

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3: 6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

… ► … ►The logarithmic integral is defined by … ►

§6.2(ii) Sine and Cosine Integrals

… ► …

4: 25.20 Approximations

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Cody et al. (1971) gives rational approximations for $\zeta\left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\leq s\leq 5$ , $5\leq s\leq 11$ , $11\leq s\leq 25$ , $25\leq s\leq 55$ . Precision is varied, with a maximum of 20S.

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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.

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Antia (1993) gives minimax rational approximations for $\Gamma\left(s+1\right)F_{s}(x)$ , where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals $-\infty<x\leq 2$ 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 $10^{-4},10^{-8},10^{-12}$ .

5: 25.19 Tables

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Abramowitz and Stegun (1964) tabulates: $\zeta\left(n\right)$ , $n=2,3,4,\dots$ , 20D (p. 811); $\operatorname{Li}_{2}\left(1-x\right)$ , $x=0(.01)0.5$ , 9D (p. 1005); $f(\theta)$ , $\theta=15^{\circ}(1^{\circ})30^{\circ}(2^{\circ})90^{\circ}(5^{\circ})180^{\circ}$ , $f(\theta)+\theta\ln\theta$ , $\theta=0(1^{\circ})15^{\circ}$ , 6D (p. 1006). Here $f(\theta)$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

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

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

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Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta\left(s\right)$ for both real and complex $s$ . §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta\left(s,a\right)$ , and §22.17 lists tables for some Dirichlet $L$ -functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

6: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

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§8.21(iii) Integral Representations

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§8.21(iv) Interrelations

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§8.21(v) Special Values

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7: 7.2 Definitions

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§7.2(ii) Dawson’s Integral

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§7.2(iii) Fresnel Integrals

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Values at Infinity

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§7.2(iv) Auxiliary Functions

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§7.2(v) Goodwin–Staton Integral

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8: 25.12 Polylogarithms

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Integral Representation

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§25.12(iii) Fermi–Dirac and Bose–Einstein Integrals

►The Fermi–Dirac and Bose–Einstein integrals are defined by … ►In terms of polylogarithms … ►For a uniform asymptotic approximation for

F_{s}(x)

see Temme and Olde Daalhuis (1990).

9: 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 Fermi–Dirac 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). …

10: 25.17 Physical Applications

… ►The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). …