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1: DLMF Project News

error generating summary

2: Foreword

… ►However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. …

3: Frank W. J. Olver

… ►Having witnessed the birth of the computer age firsthand (as a colleague of Alan Turing at NPL, for example), Olver is also well known for his contributions to the development and analysis of numerical methods for computing special functions. … ►He continued his editing work until the time of his death on April 22, 2013 at age 88.

4: Publications

… ►

A Special Functions Handbook for the Digital Age, Notices of the American Mathematical Society 58, 7 (2011), pp. 905–911.

…

5: 4.21 Identities

… ►

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

… ►

4.21.10 $$\tan u\pm \tan v=\frac{\sin \left(u\pm v\right)}{\cos u\cos v},$$

ⓘ

Symbols:

$\cos z$: cosine function, $\sin z$: sine function and $\tan z$: tangent function

A&S Ref:

4.3.38

Permalink:

http://dlmf.nist.gov/4.21.E10

Encodings:

pMML, png, TeX

See also:

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

… ►

4.21.15 $$2\sin u\sin v=\cos \left(u-v\right)-\cos \left(u+v\right),$$

ⓘ

Symbols:

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

A&S Ref:

4.3.31

Permalink:

http://dlmf.nist.gov/4.21.E15

Encodings:

pMML, png, TeX

See also:

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

… ►

4.21.17 $$2\sin u\cos v=\sin \left(u-v\right)+\sin \left(u+v\right).$$

ⓘ

Symbols:

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

A&S Ref:

4.3.33

Permalink:

http://dlmf.nist.gov/4.21.E17

Encodings:

pMML, png, TeX

See also:

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

… ►If $t=\tan \left(\frac{1}{2}z\right)$, then …

6: 4.35 Identities

… ►

4.35.14 $$2\sinh u\sinh v=\cosh \left(u+v\right)-\cosh \left(u-v\right),$$

ⓘ

Symbols:

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

A&S Ref:

4.5.38

Permalink:

http://dlmf.nist.gov/4.35.E14

Encodings:

pMML, png, TeX

See also:

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

►

4.35.15 $$2\cosh u\cosh v=\cosh \left(u+v\right)+\cosh \left(u-v\right),$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function

A&S Ref:

4.5.39

Permalink:

http://dlmf.nist.gov/4.35.E15

Encodings:

pMML, png, TeX

See also:

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

►

4.35.16 $$2\sinh u\cosh v=\sinh \left(u+v\right)+\sinh \left(u-v\right).$$

ⓘ

Symbols:

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

A&S Ref:

4.5.40

Permalink:

http://dlmf.nist.gov/4.35.E16

Encodings:

pMML, png, TeX

See also:

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

… ►

4.35.34 $$\sinh z=\sinh x\cos y+\mathrm{i}\cosh x\sin y,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.5.49

Permalink:

http://dlmf.nist.gov/4.35.E34

Encodings:

pMML, png, TeX

See also:

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

►

4.35.35 $$\cosh z=\cosh x\cos y+\mathrm{i}\sinh x\sin y,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.5.50

Permalink:

http://dlmf.nist.gov/4.35.E35

Encodings:

pMML, png, TeX

See also:

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

…

7: 4.47 Approximations

… ►Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arctan$, $\mathrm{arcsinh}$. Schonfelder (1980) gives 40D coefficients for $\sin$, $\cos$, $\tan$. … ►Hart et al. (1968) give $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arccos$, $\arctan$, $\sinh$, $\cosh$, $\tanh$, $\mathrm{arcsinh}$, $\mathrm{arccosh}$. … ►Luke (1975, Chapter 3) supplies real and complex approximations for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\arctan$, $\mathrm{arcsinh}$. …

8: 4.14 Definitions and Periodicity

… ►

4.14.4 $$\tan z=\frac{\sin z}{\cos z},$$

ⓘ

Defines:

$\tan z$: tangent function

Symbols:

$\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.3

Referenced by:

§4.14, §4.45(i)

Permalink:

http://dlmf.nist.gov/4.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

… ►

4.14.7 $$\cot z=\frac{\cos z}{\sin z}=\frac{1}{\tan z}.$$

ⓘ

Defines:

$\cot z$: cotangent function

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.6

Referenced by:

§4.14, §4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

►The functions $\sin z$ and $\cos z$ are entire. In $ℂ$ the zeros of $\sin z$ are $z=k\pi$, $k\in \mathbb{Z}$; the zeros of $\cos z$ are $z=\left(k+\frac{1}{2}\right)\pi$, $k\in \mathbb{Z}$. The functions $\tan z$, $\csc z$, $\sec z$, and $\cot z$ are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). …

9: 4.42 Solution of Triangles

… ►

4.42.8 $$\cos a=\cos b\cos c+\sin b\sin c\cos A,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $a$: arc length, $b$: arc length and $c$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

►

4.42.9 $$\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c},$$

ⓘ

Symbols:

$\sin z$: sine function, $A$: angle, $B$: angle, $C$: angle, $a$: arc length, $b$: arc length and $c$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

►

4.42.10 $$\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $B$: angle, $a$: arc length, $b$: arc length and $c$: arc length

Permalink:

http://dlmf.nist.gov/4.42.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

►

4.42.11 $$\cos a\cos C=\sin a\cot b-\sin C\cot B,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cot z$: cotangent function, $\sin z$: sine function, $B$: angle, $C$: angle, $a$: arc length and $b$: arc length

Permalink:

http://dlmf.nist.gov/4.42.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

►

4.42.12 $$\cos A=-\cos B\cos C+\sin B\sin C\cos a.$$

ⓘ

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $B$: angle, $C$: angle and $a$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

…

10: 4.28 Definitions and Periodicity

… ►

4.28.9 $$\cos \left(\mathrm{i}z\right)=\cosh z,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.5.8

Permalink:

http://dlmf.nist.gov/4.28.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.28, §4.28 and Ch.4

… ►

4.28.11 $$\csc \left(\mathrm{i}z\right)=-\mathrm{i}\mathrm{csch}z,$$

ⓘ

Symbols:

$\csc z$: cosecant function, $\mathrm{csch}z$: hyperbolic cosecant function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.5.10

Permalink:

http://dlmf.nist.gov/4.28.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.28, §4.28 and Ch.4

►

4.28.12 $$\sec \left(\mathrm{i}z\right)=\mathrm{sech}z,$$

ⓘ

Symbols:

$\mathrm{sech}z$: hyperbolic secant function, $\mathrm{i}$: imaginary unit, $\sec z$: secant function and $z$: complex variable

A&S Ref:

4.5.11

Permalink:

http://dlmf.nist.gov/4.28.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.28, §4.28 and Ch.4

… ►The functions $\sinh z$ and $\cosh z$ have period $2\pi \mathrm{i}$, and $\tanh z$ has period $\pi \mathrm{i}$. The zeros of $\sinh z$ and $\cosh z$ are $z=\mathrm{i}k\pi$ and $z=\mathrm{i}\left(k+\frac{1}{2}\right)\pi$, respectively, $k\in \mathbb{Z}$.