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1: 26.21 Tables

… ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. … ►Goldberg et al. (1976) contains tables of binomial coefficients to $n=100$ and Stirling numbers to $n=40$.

2: 4.21 Identities

§4.21 Identities

… ►

4.21.1 $$\sin u\pm \cos u=\sqrt{2}\sin \left(u\pm \frac{1}{4}\pi \right)=\pm \sqrt{2}\cos \left(u\mp \frac{1}{4}\pi \right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $\sin z$: sine function

Referenced by:

§4.21(i), Erratum (V1.0.7) for Equation (4.21.1)

Permalink:

http://dlmf.nist.gov/4.21.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.0.7):

Originally the symbol $\pm$ was missing after the second equal sign.

Suggested 2012-09-27 by Dennis M. Heim

See also:

Annotations for §4.21(i), §4.21 and Ch.4

… ►

4.21.2 $$\sin \left(u\pm v\right)=\sin u\cos v\pm \cos u\sin v,$$

ⓘ

Symbols:

$\cos z$: cosine function and $\sin z$: sine function

A&S Ref:

4.3.16

Referenced by:

§4.21(i), (9.8.13)

Permalink:

http://dlmf.nist.gov/4.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(i), §4.21 and Ch.4

… ►

4.21.4 $$\tan \left(u\pm v\right)=\frac{\tan u\pm \tan v}{1\mp \tan u\tan v},$$

ⓘ

Symbols:

$\tan z$: tangent function

A&S Ref:

4.3.18

Referenced by:

(9.8.17)

Permalink:

http://dlmf.nist.gov/4.21.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(i), §4.21 and Ch.4

►

4.21.5 $$\cot \left(u\pm v\right)=\frac{\pm \cot u\cot v-1}{\cot u\pm \cot v}.$$

ⓘ

Symbols:

$\cot z$: cotangent function

A&S Ref:

4.3.19

Permalink:

http://dlmf.nist.gov/4.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(i), §4.21 and Ch.4

…

3: 4.35 Identities

§4.35 Identities

… ►

4.35.1 $$\sinh \left(u\pm v\right)=\sinh u\cosh v\pm \cosh u\sinh v,$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function and $\sinh z$: hyperbolic sine function

A&S Ref:

4.5.24 (modified)

Permalink:

http://dlmf.nist.gov/4.35.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(i), §4.35 and Ch.4

►

4.35.2 $$\cosh \left(u\pm v\right)=\cosh u\cosh v\pm \sinh u\sinh v,$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function and $\sinh z$: hyperbolic sine function

A&S Ref:

4.5.25 (modified)

Permalink:

http://dlmf.nist.gov/4.35.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(i), §4.35 and Ch.4

►

4.35.3 $$\tanh \left(u\pm v\right)=\frac{\tanh u\pm \tanh v}{1\pm \tanh u\tanh v},$$

ⓘ

Symbols:

$\tanh z$: hyperbolic tangent function

A&S Ref:

4.5.26 (modified)

Permalink:

http://dlmf.nist.gov/4.35.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(i), §4.35 and Ch.4

►

4.35.4 $$\coth \left(u\pm v\right)=\frac{\pm \coth u\coth v+1}{\coth u\pm \coth v}.$$

ⓘ

Symbols:

$\coth z$: hyperbolic cotangent function

A&S Ref:

4.5.27 (modified)

Permalink:

http://dlmf.nist.gov/4.35.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(i), §4.35 and Ch.4

…

4: 12.19 Tables

… ►

Abramowitz and Stegun (1964, Chapter 19) includes $U(a,x)$ and $V(a,x)$ for $\pm a=0(.1)1(.5)5$, $x=0(.1)5$, 5S; $W(a,\pm x)$ for $\pm a=0(.1)1(1)5$, $x=0(.1)5$, 4-5D or 4-5S.

… ►

Kireyeva and Karpov (1961) includes $D_p\left(x(1+\mathrm{i})\right)$ for $\pm x=0(.1)5$, $p=0(.1)2$, and $\pm x=5(.01)10$, $p=0(.5)2$, 7D.

►

Karpov and Čistova (1964) includes $D_p\left(x\right)$ for $p=-2(.1)0$, $\pm x=0(.01)5$; $p=-2(.05)0$, $\pm x=5(.01)10$, 6D.

… ►

Zhang and Jin (1996, pp. 455–473) includes $U(\pm n-\frac{1}{2},x)$, $V(\pm n-\frac{1}{2},x)$, $U(\pm \nu -\frac{1}{2},x)$, $V(\pm \nu -\frac{1}{2},x)$, and derivatives, $\nu =n+\frac{1}{2}$, $n=0(1)10(10)30$, $x=0.5,1,5,10,30,50$, 8S; $W(a,\pm x)$, $W(-a,\pm x)$, and derivatives, $a=h(1)5+h$, $x=0.5,1$ and $a=h(1)5+h$, $x=5$, $h=0,0.5$, 8S. Also, first zeros of $U(a,x)$, $V(a,x)$, and of derivatives, $a=-6(.5)-1$, 6D; first three zeros of $W(a,-x)$ and of derivative, $a=0(.5)4$, 6D; first three zeros of $W(-a,\pm x)$ and of derivative, $a=0.5(.5)5.5$, 6D; real and imaginary parts of $U(a,z)$, $a=-1.5(1)1.5$, $z=x+\mathrm{i}y$, $x=0.5,1,5,10$, $y=0(.5)10$, 8S.

…

5: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

►For properties of Struve functions of argument $x{\mathrm{e}}^{\pm 3\pi \mathrm{i}/4}$ see McLachlan and Meyers (1936).

6: 4.16 Elementary Properties

… ►

Table 4.16.2: Trigonometric functions: quarter periods and change of sign.

► ►►►►►►

$x$	$-\theta$	$\frac{1}{2}\pi \pm \theta$	$\pi \pm \theta$	$\frac{3}{2}\pi \pm \theta$	$2\pi \pm \theta$
$\sin x$	$-\sin \theta$	$\cos \theta$	$\mp \sin \theta$	$-\cos \theta$	$\pm \sin \theta$
$\cos x$	$\cos \theta$	$\mp \sin \theta$	$-\cos \theta$	$\pm \sin \theta$	$\cos \theta$
$\tan x$	$-\tan \theta$	$\mp \cot \theta$	$\pm \tan \theta$	$\mp \cot \theta$	$\pm \tan \theta$
…
$\cot x$	$-\cot \theta$	$\mp \tan \theta$	$\pm \cot \theta$	$\mp \tan \theta$	$\pm \cot \theta$

►

…

7: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

… ►

28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.1 (for 3 and for integer $\nu$) 20.9.3 (for 4 and for integer $\nu$)

Referenced by:

§28.25

Permalink:

http://dlmf.nist.gov/28.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

… ►

$D_{-1}^{\pm }=0,$

►

$D_0^{\pm }=1,$

… ►

28.25.3 $$(m+1)D_{m+1}^{\pm }+\left({(m+\frac{1}{2})}^2\pm (m+\frac{1}{4})8\mathrm{i}h+2h^2-a\right)D_m^{\pm }\pm (m-\frac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1}^{\pm }=0,$$ $m\ge 0$.

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, $m$: integer, $h$: parameter, $a$: parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.2 (for $+$) 20.9.4 (for $-$)

Permalink:

http://dlmf.nist.gov/28.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

…

8: 4.13 Lambert $W$-Function

… ►The decreasing solution can be identified as $W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$. …$W_0\left(z\right)$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. …The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. … ►and has several advantages over the Lambert $W$-function (see Lawrence et al. (2012)), and the tree $T$-function $T\left(z\right)=-W\left(-z\right)$, which is a solution of … ►where $t\ge 0$ for $W_0$, $t\le 0$ for $W_{\pm 1}$ on the relevant branch cuts, …

9: 36.7 Zeros

… ►

$$x_{m,n}^{\pm }=\sqrt{\frac{2}{-y_m}}\left(2n+\frac{1}{2}+{(-1)}^m\frac{1}{2}\pm \frac{1}{4}\right)\pi ,$$ $m=1,2,3,\mathrm{\dots }$, $n=0,\pm 1,\pm 2,\mathrm{\dots }$.

►

Table 36.7.1: Zeros of cusp diffraction catastrophe to 5D. …

► ►►►►►

Zeros $\left\{\genfrac{}{}{0.0pt}{}{x}{y}\right\}$ inside, and zeros $\left[\genfrac{}{}{0.0pt}{}{x}{y}\right]$ outside, the cusp $x^2=\frac{8}{27}{|y|}^3$.
$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 0.52768}{-4.37804}}\right\}$	$\left[{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 2.35218}{-1.74360}}\right]$
$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 1.41101}{-5.55470}}\right\}$	$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 2.36094}{-5.52321}}\right\}$	$\left[{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 4.42707}{-3.05791}}\right]$
…
$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 0.38488}{-8.31916}}\right\}$	$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 2.71193}{-8.22315}}\right\}$	$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 3.49286}{-8.20326}}\right\}$	$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 5.96669}{-7.85723}}\right\}$	$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 6.79538}{-7.80456}}\right\}$	$\left[{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 9.17308}{-5.55831}}\right]$

►

… ►

$x_n=\pm {\left({\displaystyle \frac{8}{27}}\right)}^{1/2}{|y_n|}^{3/2}(1+{\xi }_n),$

…

10: Charles W. Clark

… ►He has been a Visiting Fellow at the Australian National University, a Dr. Lee Fellow at Christ Church College of the University of Oxford, and Visiting Professor at the National University of Singapore. ►Clark received the R&D 100 Award, Distinguished Presidential Rank Award of the U. …He has served as Chair of the Division of Atomic, Molecular, and Optical Physics of the APS, Chair of the Physics Section of the AAAS, and as Program Manager for Atomic, Molecular, and Quantum Physics at the U. …