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11: 25.12 Polylogarithms

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Figure 25.12.1: Dilogarithm function ${\mathrm{Li}}_2\left(x\right)$, Magnify

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Figure 25.12.2: Absolute value of the dilogarithm function $\left|{\mathrm{Li}}_2\left(x+\mathrm{i}y\right)\right|$, $-20\le x\le 20$, $-20\le y\le 20$. … Magnify 3D Help

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25.12.11 $${\mathrm{Li}}_s\left(z\right)\equiv \frac{z}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}}{{\mathrm{e}}^x-z}dx,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\int$: integral, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $x$: real variable, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Lewin (1981, (7.188), p. 236)

Referenced by:

(25.12.16), §25.12(ii)

Permalink:

http://dlmf.nist.gov/25.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

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25.12.14 $$F_s(x)=\frac{1}{\mathrm{\Gamma }\left(s+1\right)}{\int }_0^{\mathrm{\infty }}\frac{t^s}{{\mathrm{e}}^{t-x}+1}dt,$$ $s>-1$,

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Defines:

$F_s(x)$: Fermi–Dirac integral (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $x$: real variable and $s$: complex variable

Keywords:

definition, Fermi–Dirac integral, improper integral, integral representation

Source:

Dingle (1957b, (1), p. 226)

Referenced by:

(25.12.16), 3rd item, 5th item

Permalink:

http://dlmf.nist.gov/25.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(iii), §25.12 and Ch.25

… ►Sometimes the factor $1/\mathrm{\Gamma }\left(s+1\right)$ is omitted. …

12: 32.8 Rational Solutions

… ►with $\kappa$, $\lambda$, and $\mu$ arbitrary constants. ►In the general case assume $\gamma \delta \ne 0$, so that as in §32.2(ii) we may set $\gamma =1$ and $\delta =-1$. … ►with $\kappa$ and $\mu$ arbitrary constants. … ►

$\alpha =\frac{1}{2}{a}^2$, $\beta =-\frac{1}{2}{(a+n)}^2$, and $\gamma =m$, with $m+n$ even.

… ►with $\kappa$ and $\mu$ arbitrary constants. …

13: 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 $0.5\le s\le 5$, $5\le s\le 11$, $11\le s\le 25$, $25\le s\le 55$. Precision is varied, with a maximum of 20S.

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

14: 30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

►As ${\gamma }^2\to +\mathrm{\infty }$, with $q=2(n-m)+1$, … ►The asymptotic behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ and $a_{n,k}^m({\gamma }^2)$ as $n\to \mathrm{\infty }$ in descending powers of $2n+1$ is derived in Meixner (1944). …The asymptotic behavior of ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ and ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$ as $x\to \pm 1$ is given in Erdélyi et al. (1955, p. 151). The behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ for complex ${\gamma }^2$ and large $|{\lambda }_n^m\left({\gamma }^2\right)|$ is investigated in Hunter and Guerrieri (1982). …

15: 8 Incomplete Gamma and Related
Functions

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16: 28 Mathieu Functions and Hill’s Equation

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17: 23 Weierstrass Elliptic and Modular
Functions

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18: 36.5 Stokes Sets

… ►where $j$ denotes a real critical point (36.4.1) or (36.4.2), and $\mu$ denotes a critical point with complex $t$ or $s,t$, connected with $j$ by a steepest-descent path (that is, a path where $\mathrm{\Re }\mathrm{\Phi }=\mathrm{constant}$) in complex $t$ or $(s,t)$ space. … ►

36.5.4 $$80x^5-40x^4-55x^3+5x^2+20x-1=0,$$

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Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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Figure 36.5.5: Elliptic umbilic catastrophe with $z=\mathrm{constant}$. … Magnify

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Figure 36.5.6: Hyperbolic umbilic catastrophe with $z=\mathrm{constant}$. Magnify

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19: 11.6 Asymptotic Expansions

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11.6.1 $${𝐊}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.29

Referenced by:

§11.6(i), §11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

►where $\delta$ is an arbitrary small positive constant. … ►

11.6.2 $${𝐌}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.2.6

Referenced by:

§11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

… ►where $\gamma$ is Euler’s constant (§5.2(ii)). … ►

$c_3(\lambda )=20{\lambda }^{-6}-4{\lambda }^{-4},$

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20: 5.11 Asymptotic Expansions

… ►The scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ is defined in (5.11.3) and its main property is ${\mathrm{\Gamma }}^{\ast }\left(z\right)\sim 1$ as $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$. Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►In this subsection $a$, $b$, and $c$ are real or complex constants. … ►

5.11.12 $$\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\sim z^{a-b},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.11(i)

Permalink:

http://dlmf.nist.gov/5.11.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

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5.11.19 $$\frac{\mathrm{\Gamma }\left(z+a\right)\mathrm{\Gamma }\left(z+b\right)}{\mathrm{\Gamma }\left(z+c\right)}\sim \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{{\left(c-a\right)}_k{\left(c-b\right)}_k}{k!}\mathrm{\Gamma }\left(a+b-c+z-k\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $k$: nonnegative integer, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.11(i), §5.11(iii)

Permalink:

http://dlmf.nist.gov/5.11.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

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