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21: Peter L. Walker

… ►was Professor of Mathematics at the American University of Sharjah, Sharjah, United Arab Emirates, in 1997–2005. He began his academic career in 1964 at the University of Lancaster, U. …Since 1984 he has also taught at other Persian Gulf universities, including Sultan Qaboos University, Oman. … ►

20 Theta Functions

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22: 4.31 Special Values and Limits

§4.31 Special Values and Limits

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Table 4.31.1: Hyperbolic functions: values at multiples of $\frac{1}{2}\pi \mathrm{i}$.

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$z$	0	$\frac{1}{2}\pi \mathrm{i}$	$\pi \mathrm{i}$	$\frac{3}{2}\pi \mathrm{i}$	$\mathrm{\infty }$
…

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23: 18.40 Methods of Computation

… ►There are many ways to implement these first two steps, noting that the expressions for ${\alpha }_n$ and ${\beta }_n$ of equation (18.2.30) are of little practical numerical value, see Gautschi (2004) and Golub and Meurant (2010). … ►Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. … ►The bottom and top of the steps at the $x_i$ are lower and upper bounds to ${\int }_a^{x_i}d\mu (x)$ as made explicit via the Chebyshev inequalities discussed by Shohat and Tamarkin (1970, pp. 42–43). … ►This is a challenging case as the desired $w^{\mathrm{RCP}}(x)$ on $[-1,1]$ has an essential singularity at $x=-1$. … ►Achieving precisions at this level shown above requires higher than normal computational precision, see Gautschi (2009). …

24: Staff

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Ronald F. Boisvert, Editor at Large, NIST

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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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25: 28.7 Analytic Continuation of Eigenvalues

§28.7 Analytic Continuation of Eigenvalues

… ►The branch points are called the exceptional values, and the other points normal values. The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). All real values of $q$ are normal values. … ► …

26: William P. Reinhardt

… ► 1942 in San Francisco, California) is Professor of Chemistry and Adjunct Professor of Physics at the University of Washington, Seattle, currently Emeritus. His undergraduate and graduate degrees are from the University of California at Berkeley and Harvard University, respectively. …Reinhardt is a frequent visitor to the NIST Physics Laboratory in Gaithersburg, and to the Joint Quantum Institute (JQI) and Institute for Physical Sciences and Technology (ISTP) at the University of Maryland. … ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

27: 4.46 Tables

… ►Extensive numerical tables of all the elementary functions for real values of their arguments appear in Abramowitz and Stegun (1964, Chapter 4). … ►For 40D values of the first 500 roots of $\tan x=x$, see Robinson (1972). (These roots are zeros of the Bessel function $J_{3/2}\left(x\right)$; see §10.21.) ►For 10S values of the first five complex roots of $\sin z=az$, $\cos z=az$, and $\cosh z=az$, for selected positive values of $a$, see Fettis (1976). …

28: 32.8 Rational Solutions

… ► ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. … ►

32.8.3 $$w(z;3)=\frac{3z^2}{z^3+4}-\frac{6z^2(z^3+10)}{z^6+20z^3-80},$$

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Symbols:

$z$: real

Permalink:

http://dlmf.nist.gov/32.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.8(ii), §32.8 and Ch.32

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$$Q_3(z)=z^6+20z^3-80,$$

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

… ►where $n\in \mathbb{Z}$, $a={\epsilon }_1\sqrt{2\alpha }$, $b={\epsilon }_2\sqrt{-2\beta }$, $c={\epsilon }_3\sqrt{2\gamma }$, and $d={\epsilon }_4\sqrt{1-2\delta }$, with ${\epsilon }_j=\pm 1$, $j=1,2,3,4$, independently, and at least one of $a$, $b$, $c$ or $d$ is an integer. …

29: 12.11 Zeros

… ►If , then $U(a,x)$ has no positive real zeros. If , $n=1,2,\mathrm{\dots }$, then $U(a,x)$ has $n$ positive real zeros. … ►If $a>-\frac{1}{2}$, then $V(a,x)$ has no positive real zeros, and if $a=\frac{3}{2}-2n$, $n\in \mathbb{Z}$, then $V(a,x)$ has a zero at $x=0$. … ►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …Here $\alpha ={\mu }^{-\frac{4}{3}}{a}_s$, $a_s$ denoting the $s$th negative zero of the function $\mathrm{Ai}$ (see §9.9(i)). …

30: 1.6 Vectors and Vector-Valued Functions

§1.6 Vectors and Vector-Valued Functions

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§1.6(iii) Vector-Valued Functions

… ►The divergence of a differentiable vector-valued function $𝐅=F_1𝐢+F_2𝐣+F_3𝐤$ is … ►

Stokes’s Theorem

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