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11: Bibliography Z

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On the Computation of Zeros of Bessel and Bessel-related Functions. In Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.), Utrecht, pp. 409–416.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

BibTeX

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Remark on “Algorithm 916: computing the Faddeyeva and Voigt functions”: efficiency improvements and Fortran translation. ACM Trans. Math. Softw. 42 (3), pp. 26:1–26:9.

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Notes:

Includes MATLAB and Fortran programs claiming 6S or 13S accuracy for single and double precision

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry

Referenced by:

3rd item, 6th item, §7.25(iii), §7.25(vi).

Encodings:

BibTeX

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Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.

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External Links:

ISSN 0024-6115, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

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Mathieu functions for purely imaginary parameters. J. Comput. Appl. Math. 236 (17), pp. 4513–4524.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (Bartolomeu D. B. Figueiredo), ZentralBlatt

Referenced by:

§28.36(iii), §28.36(iii).

Encodings:

BibTeX

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Fast evaluation of elementary mathematical functions with correctly rounded last bit. ACM Trans. Math. Software 17 (3), pp. 410–423.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§4.45(i).

Encodings:

BibTeX

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12: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols, different asymptotic forms are obtained depending on which parameters are large. … ►and the symbol $o\left(1\right)$ denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i). … ►Uniform approximations in terms of Airy functions for the $\mathit{3}j$ and $\mathit{6}j$ symbols are given in Schulten and Gordon (1975b). For approximations for the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.

13: 30.16 Methods of Computation

… ►For $d$ sufficiently large, construct the $d\times d$ tridiagonal matrix $𝐀=[A_{j,k}]$ with nonzero elements …and real eigenvalues ${\alpha }_{1,d}$, ${\alpha }_{2,d}$, $\mathrm{\dots }$, ${\alpha }_{d,d}$, arranged in ascending order of magnitude. … ►Let $𝐀$ be the $d\times d$ matrix given by (30.16.1) if $n-m$ is even, or by (30.16.6) if $n-m$ is odd. Form the eigenvector ${[e_{1,d},e_{2,d},\mathrm{\dots },e_{d,d}]}^{\mathrm{T}}$ of $𝐀$ associated with the eigenvalue ${\alpha }_{p,d}$, $p=\lfloor \frac{1}{2}(n-m)\rfloor +1$, normalized according to … ►The coefficients $a_{n,k}^m({\gamma }^2)$ calculated in §30.16(ii) can be used to compute $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$ from (30.11.3) as well as the connection coefficients $K_n^m(\gamma )$ from (30.11.10) and (30.11.11). …

14: Bibliography S

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Numerical evaluation of integrals of the form $I={\int }_{x_1}^{x_2}f(x)e^{i\varphi (x)}𝑑x$ and the tabulation of the function $\mathrm{Gi}(z)=(1/\pi ){\int }_0^{\mathrm{\infty }}\sin (uz+\frac{1}{3}{u}^3)𝑑u$ . Quart. J. Mech. Appl. Math. 3 (1), pp. 107–112.

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External Links:

Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent. Phys. D 3 (1-2), pp. 165–184.

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External Links:

Document, ZentralBlatt

Referenced by:

§32.11(ii).

Encodings:

BibTeX

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Inequalities involving cylindrical functions of nearly equal argument and order. Proc. Amer. Math. Soc. 5 (3), pp. 337–344.

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External Links:

ISSN 0002-9939, FullText, MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

§10.14.

Encodings:

BibTeX

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Some comments on Fourier analysis, uncertainty and modeling. SIAM Rev. 25 (3), pp. 379–393.

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External Links:

ISSN 0036-1445, Document, MathReview (H. Vincent Poor), ZentralBlatt

Referenced by:

§30.15(ii), §30.15(iii), §30.15(iv), §30.15(v), §30.15(v).

Encodings:

BibTeX

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On the relative extrema of ultraspherical polynomials. Boll. Un. Mat. Ital. (3) 5, pp. 125–127.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.14(iii).

Encodings:

BibTeX

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15: 34.14 Tables

§34.14 Tables

►Tables of exact values of the squares of the $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols on pp. … ►Tables of $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 17/2$ are given in Appel (1968) to 6D. …Other tabulations for $\mathit{3}j$ symbols are listed on pp. … ►In Varshalovich et al. (1988) algebraic expressions for the Clebsch–Gordan coefficients with all parameters $\le 5$ and numerical values for all parameters $\le 3$ are given on pp. …

16: 9.13 Generalized Airy Functions

… ►where $m=3,4,5,\mathrm{\dots }.$ For real variables the solutions of (9.13.13) are denoted by $U_m(t)$, $U_m(-t)$ when $m$ is even, and by $V_m(t)$, ${\overline{V}}_m(t)$ when $m$ is odd. … ►

9.13.27 $$B_k(z,p)={\int }_{{ℐ}_k}{t}^{-p}\exp \left(zt-\frac{1}{3}{t}^3\right)dt,$$ $k=1,2,3$, $p=0,\pm 1,\pm 2,\mathrm{\dots }$,

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Symbols:

$B_k(z,p)$: generalized Airy function, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $z$: complex variable, $k$: index, $p$: parameter and $ℐ$: integration path

Source:

Drazin and Reid (1981, (A20), p. 470)

Permalink:

http://dlmf.nist.gov/9.13.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(ii), §9.13 and Ch.9

… ►The integration paths ${ℒ}_0$, ${ℒ}_1$, ${ℒ}_2$, ${ℒ}_3$ are depicted in Figure 9.13.1. ${ℐ}_1$, ${ℐ}_2$, ${ℐ}_3$ are depicted in Figure 9.13.2. … ►

9.13.31 $$\frac{d^{3}w}{{dz}^3}-z\frac{dw}{dz}+(p-1)w=0,$$

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Defines:

$w$: function (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $p$: parameter

Referenced by:

§9.13(ii)

Permalink:

http://dlmf.nist.gov/9.13.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(ii), §9.13 and Ch.9

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17: 28.8 Asymptotic Expansions for Large $q$

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28.8.1 $$\begin{array}{c}{a}_m\left(h^2\right)\\ b_{m+1}\left(h^2\right)\end{array}\}\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation, $b_n\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $m$: integer and $h$: parameter

A&S Ref:

20.2.30 (in different form)

Referenced by:

§28.8(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)

Permalink:

http://dlmf.nist.gov/28.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(i), §28.8 and Ch.28

►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). … ►

28.8.2 $$b_{m+1}\left(h^2\right)-a_m\left(h^2\right)=\frac{2^{4m+5}}{m!}{\left(\frac{2}{\pi }\right)}^{1/2}{h}^{m+(3/2)}{\mathrm{e}}^{-4h}\left(1-\frac{6m^2+14m+7}{32h}+O\left(\frac{1}{h^2}\right)\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $b_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $m$: integer and $h$: parameter

A&S Ref:

20.2.31

Permalink:

http://dlmf.nist.gov/28.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(i), §28.8 and Ch.28

… ►Also let $\xi =2\sqrt{h}\cos x$ and $D_m\left(\xi \right)={\mathrm{e}}^{-{\xi }^2/4}{\mathrm{𝐻𝑒}}_m\left(\xi \right)$ (§18.3). … ►

${\sigma }_m\sim 1+{\displaystyle \frac{s}{2^{3}h}}+{\displaystyle \frac{4s^2+3}{2^7{h}^2}}+{\displaystyle \frac{19s^3+59s}{2^{11}{h}^3}}+\mathrm{\cdots },$

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18: 32.11 Asymptotic Approximations for Real Variables

… ►and $d$ and ${\theta }_0$ are constants. … ►

If , then $w(x)$ changes sign once, from positive to negative, as $x$ passes from $x_0$ to $0$.

… ►Connection formulas for $d$ and ${\theta }_0$ are given by … ►Connection formulas for $d$ and ${\theta }_0$ are given by … ►In terms of the parameter $k$ that is used in these figures $h=2^{3/2}{k}^2$. …

19: 25.16 Mathematical Applications

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25.16.6 $$H\left(s\right)=-{\zeta }^{\prime }\left(s\right)+\gamma \zeta \left(s\right)+\frac{1}{2}\zeta \left(s+1\right)+\sum _{r=1}^k\zeta \left(1-2r\right)\zeta \left(s+2r\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{2k+2}}dx,$$

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Symbols:

$\gamma$: Euler’s constant, $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler sum, finite sum, improper integral, infinite series, integral representation, series representation

Source:

Apostol and Vu (1984, (3), p. 87)

Permalink:

http://dlmf.nist.gov/25.16.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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$H\left(2\right)=2\zeta \left(3\right),$

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$H\left(3\right)=\frac{5}{4}\zeta \left(4\right),$

… ► $H\left(s\right)$ has a simple pole with residue $\zeta \left(1-2r\right)$ ($=-B_{2r}/(2r)$) at each odd negative integer $s=1-2r$, $r=1,2,3,\mathrm{\dots }$. … ►

25.16.15 $$\sum _{r=1}^{\mathrm{\infty }}\sum _{k=1}^r\frac{1}{r^2(r+k)}=\frac{3}{4}\zeta \left(3\right).$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function and $k$: nonnegative integer

Keywords:

infinite series, special value

Source:

Apostol and Vu (1984, (15), p. 94)

Permalink:

http://dlmf.nist.gov/25.16.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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20: 19.27 Asymptotic Approximations and Expansions

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§19.27(iv) $R_D(x,y,z)$

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19.27.7 $$R_D(x,y,z)=\frac{3}{2z^{3/2}}\left(\ln \left(\frac{8z}{a+g}\right)-2\right)\left(1+O\left(\frac{a}{z}\right)\right),$$ $a/z\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.27.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

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19.27.8 $$R_D(x,y,z)=\frac{3}{\sqrt{xyz}}-\frac{6}{xy}{R}_G(x,y,0)\left(1+O\left(\frac{z}{g}\right)\right),$$ $z/g\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $g$

Referenced by:

§19.20(iv), §19.29(iii)

Permalink:

http://dlmf.nist.gov/19.27.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

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19.27.9 $$R_D(x,y,z)=\frac{3}{\sqrt{xz}(\sqrt{y}+\sqrt{z})}\left(1+O\left(\frac{b}{x}\ln \frac{x}{b}\right)\right),$$ $b/x\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\ln z$: principal branch of logarithm function and $b$

Permalink:

http://dlmf.nist.gov/19.27.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

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19.27.10 $$R_D(x,y,z)=R_D(0,y,z)-\frac{3\sqrt{x}}{hz}\left(1+O\left(\sqrt{\frac{x}{h}}\right)\right),$$ $x/h\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $h$

Permalink:

http://dlmf.nist.gov/19.27.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

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