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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: 22.9 Cyclic Identities

§22.9 Cyclic Identities

… ►

§22.9(ii) Typical Identities of Rank 2

… ► ►

§22.9(iii) Typical Identities of Rank 3

… ►