classification%20of%20parameters

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21: Peter L. Walker

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20 Theta Functions

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22: Staff

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William P. Reinhardt, University of Washington, Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23

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William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23

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23: 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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25.12.13 $${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}a}\right)+{\mathrm{e}}^{\pi \mathrm{i}s}{\mathrm{Li}}_s\left({\mathrm{e}}^{-2\pi \mathrm{i}a}\right)=\frac{{(2\pi )}^s{\mathrm{e}}^{\pi \mathrm{i}s/2}}{\mathrm{\Gamma }\left(s\right)}\zeta (1-s,a),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $(a,b)$: open interval, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Source:

Erdélyi et al. (1953a, (1.11.16), p. 31); with ${\mathrm{Li}}_s\left(z\right)=F(z,s)$, $z={\mathrm{e}}^{2\pi \mathrm{i}a}$

Referenced by:

§25.12(ii)

Permalink:

http://dlmf.nist.gov/25.12.E13

Encodings:

pMML, png, TeX

See also:

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

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24: 36.5 Stokes Sets

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36.5.2 $$y^3=\frac{27}{4}\left(\sqrt{27}-5\right)x^2=1.32403x^2.$$

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Symbols:

$y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.5.E2

Encodings:

pMML, png, TeX

See also:

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

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

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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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36.5.11 $$\frac{x}{z^2}=-1-12u^2+8u-\left|\frac{y}{z^2}\right|\frac{\frac{1}{3}-u}{{\left(u\left(\frac{2}{3}-u\right)\right)}^{1/2}}.$$

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Symbols:

$y$: real parameter, $z$: real parameter and $x$: real parameter

Referenced by:

§36.5(iv)

Permalink:

http://dlmf.nist.gov/36.5.E11

Encodings:

pMML, png, TeX

See also:

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

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36.5.12 $$8u^3-4u^2-\left|\frac{y}{3z^2}\right|{\left(\frac{u}{\frac{2}{3}-u}\right)}^{1/2}=\frac{y^2}{6w{z}^4}-2w^3-2w^2,$$

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Symbols:

$y$: real parameter, $z$: real parameter and $w$: quantity

Permalink:

http://dlmf.nist.gov/36.5.E12

Encodings:

pMML, png, TeX

See also:

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

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25: 10.75 Tables

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Achenbach (1986) tabulates $J_0\left(x\right)$, $J_1\left(x\right)$, $Y_0\left(x\right)$, $Y_1\left(x\right)$, $x=0(.1)8$, 20D or 18–20S.

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Bickley et al. (1952) tabulates $x^{-n}{I}_n\left(x\right)$ or ${\mathrm{e}}^{-x}{I}_n\left(x\right)$, $x^n{K}_n\left(x\right)$ or ${\mathrm{e}}^x{K}_n\left(x\right)$, $n=2(1)20$, $x=0$(.01 or .1) 10(.1) 20, 8S; $I_n\left(x\right)$, $K_n\left(x\right)$, $n=0(1)20$, $x=0$ or $0.1(.1)20$, 10S.

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Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_n\left(z\right)$ and $K_n^{\prime }\left(z\right)$, for $n=2(1)20$, 9S.

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Zhang and Jin (1996, p. 322) tabulates $\mathrm{ber}x$, ${\mathrm{ber}}^{\prime }x$, $\mathrm{bei}x$, ${\mathrm{bei}}^{\prime }x$, $\ker x$, ${\ker }^{\prime }x$, $\mathrm{kei}x$, ${\mathrm{kei}}^{\prime }x$, $x=0(1)20$, 7S.

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Zhang and Jin (1996, p. 323) tabulates the first $20$ real zeros of $\mathrm{ber}x$, ${\mathrm{ber}}^{\prime }x$, $\mathrm{bei}x$, ${\mathrm{bei}}^{\prime }x$, $\ker x$, ${\ker }^{\prime }x$, $\mathrm{kei}x$, ${\mathrm{kei}}^{\prime }x$, 8D.

26: 36.4 Bifurcation Sets

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36.4.5 $$x=0.$$

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Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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36.4.6 $$27x^2=-8y^3.$$

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Symbols:

$y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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$x=\frac{9}{20}{z}^2.$

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$x=-\frac{3}{20}{z}^2,$

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36.4.13 $$x=y=-\frac{1}{4}{z}^2.$$

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Symbols:

$y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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27: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. …

28: 2.8 Differential Equations with a Parameter

§2.8 Differential Equations with a Parameter

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§2.8(i) Classification of Cases

… ►The parameter $u$ is assumed to be real and positive. … ►

§2.8(vi) Coalescing Transition Points

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29: 28.1 Special Notation

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$m,n$	integers.
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$\nu$	order of the Mathieu function or modified Mathieu function. (When $\nu$ is an integer it is often replaced by $n$.)
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$a,q,h$	real or complex parameters of Mathieu’s equation with $q=h^2$.
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… ►Alternative notations for the parameters $a$ and $q$ are shown in Table 28.1.1. ►

Table 28.1.1: Notations for parameters in Mathieu’s equation.

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Reference	$a$	$q$
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Abramowitz and Stegun (1964, Chapter 20)

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30: 16.4 Argument Unity

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§16.4(i) Classification

… ►The last condition is equivalent to the sum of the top parameters plus $2$ equals the sum of the bottom parameters, that is, the series is 2-balanced. … ►The function ${}_3{}^{}F_2^{}(a,b,c;d,e;1)$ is analytic in the parameters $a,b,c,d,e$ when its series expansion converges and the bottom parameters are not negative integers or zero. … ►These series contain $\mathit{6}j$ symbols as special cases when the parameters are integers; compare §34.4. … ►Contiguous balanced series have parameters shifted by an integer but still balanced. …