Cosines

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1: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

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§8.21(iv) Interrelations

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§8.21(v) Special Values

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§8.21(viii) Asymptotic Expansions

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2: 6.2 Definitions and Interrelations

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§6.2(ii) Sine and Cosine Integrals

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Values at Infinity

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Hyperbolic Analogs of the Sine and Cosine Integrals

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§6.2(iii) Auxiliary Functions

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3: 4.35 Identities

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4.35.7 $$\cosh u+\cosh v=2\cosh \left(\frac{u+v}{2}\right)\cosh \left(\frac{u-v}{2}\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function

A&S Ref:

4.5.43

Permalink:

http://dlmf.nist.gov/4.35.E7

Encodings:

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See also:

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

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

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Symbols:

$\cosh z$: hyperbolic cosine function

A&S Ref:

4.5.39

Permalink:

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

Encodings:

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

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4.35.24 $$\cosh \left(-z\right)=\cosh z,$$

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Symbols:

$\cosh z$: hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.5.22

Permalink:

http://dlmf.nist.gov/4.35.E24

Encodings:

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

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4.35.30 $$\cosh \left(3z\right)=-3\cosh z+4{\cosh }^{3}z,$$

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Symbols:

$\cosh z$: hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.5.35

Permalink:

http://dlmf.nist.gov/4.35.E30

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

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

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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

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

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4: 6.17 Physical Applications

§6.17 Physical Applications

… ►Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.

5: 4.21 Identities

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4.21.8 $$\cos u+\cos v=2\cos \left(\frac{u+v}{2}\right)\cos \left(\frac{u-v}{2}\right),$$

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Symbols:

$\cos z$: cosine function

A&S Ref:

4.3.36

Permalink:

http://dlmf.nist.gov/4.21.E8

Encodings:

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See also:

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

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

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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:

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

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4.21.16 $$2\cos u\cos v=\cos \left(u-v\right)+\cos \left(u+v\right),$$

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Symbols:

$\cos z$: cosine function

A&S Ref:

4.3.32

Permalink:

http://dlmf.nist.gov/4.21.E16

Encodings:

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

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4.21.31 $$\cos \left(3z\right)=-3\cos z+4{\cos }^{3}z,$$

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Symbols:

$\cos z$: cosine function and $z$: complex variable

A&S Ref:

4.3.28

Permalink:

http://dlmf.nist.gov/4.21.E31

Encodings:

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

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4.21.33 $$\cos \left(4z\right)=8{\cos }^{4}z-8{\cos }^{2}z+1.$$

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Symbols:

$\cos z$: cosine function and $z$: complex variable

A&S Ref:

4.3.30

Permalink:

http://dlmf.nist.gov/4.21.E33

Encodings:

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See also:

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

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6: 6.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the argument. ►The main functions treated in this chapter are the exponential integrals $\mathrm{Ei}\left(x\right)$, $E_1\left(z\right)$, and $\mathrm{Ein}\left(z\right)$; the logarithmic integral $\mathrm{li}\left(x\right)$; the sine integrals $\mathrm{Si}\left(z\right)$ and $\mathrm{si}\left(z\right)$; the cosine integrals $\mathrm{Ci}\left(z\right)$ and $\mathrm{Cin}\left(z\right)$.

7: 6.4 Analytic Continuation

§6.4 Analytic Continuation

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6.4.4 $$\mathrm{Ci}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Ci}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E4

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Annotations for §6.4 and Ch.6

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6.4.7 $$\mathrm{g}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\mp \pi \mathrm{i}{\mathrm{e}}^{\mp \mathrm{i}z}+\mathrm{g}\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.12(ii), §6.4

Permalink:

http://dlmf.nist.gov/6.4.E7

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Annotations for §6.4 and Ch.6

►Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $E_1\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.

8: 4.14 Definitions and Periodicity

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4.14.2 $$\cos z=\frac{{\mathrm{e}}^{\mathrm{i}z}+{\mathrm{e}}^{-\mathrm{i}z}}{2},$$

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Defines:

$\cos z$: cosine function

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.3.2

Referenced by:

(9.5.3)

Permalink:

http://dlmf.nist.gov/4.14.E2

Encodings:

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See also:

Annotations for §4.14 and Ch.4

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4.14.6 $$\sec z=\frac{1}{\cos z},$$

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Defines:

$\sec z$: secant function

Symbols:

$\cos z$: cosine function and $z$: complex variable

A&S Ref:

4.3.5

Permalink:

http://dlmf.nist.gov/4.14.E6

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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 $z=\left(k+\frac{1}{2}\right)\pi$, $k\in \mathbb{Z}$. … ►

4.14.9 $$\cos \left(z+2k\pi \right)=\cos z,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $k$: integer and $z$: complex variable

A&S Ref:

4.3.8

Permalink:

http://dlmf.nist.gov/4.14.E9

Encodings:

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See also:

Annotations for §4.14 and Ch.4

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9: 4.32 Inequalities

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4.32.1 $$\cosh x\le {\left(\frac{\sinh x}{x}\right)}^3,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E1

Encodings:

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See also:

Annotations for §4.32 and Ch.4

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4.32.2 $x>0$,

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Symbols:

$\cos z$: cosine function, $\tanh z$: hyperbolic tangent function, $\sin z$: sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E2

Encodings:

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See also:

Annotations for §4.32 and Ch.4

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4.32.3 $$|\cosh x-\cosh y|\ge |x-y|\sqrt{\sinh x\sinh y},$$ $x>0$, $y>0$,

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E3

Encodings:

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See also:

Annotations for §4.32 and Ch.4

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10: 4.42 Solution of Triangles

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4.42.5 $$c^2=a^2+b^2-2ab\cos C,$$

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Symbols:

$\cos z$: cosine function, $a$: length, $b$: base, $c$: length and $C$: angle

A&S Ref:

4.3.148

Permalink:

http://dlmf.nist.gov/4.42.E5

Encodings:

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See also:

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

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4.42.6 $$a=b\cos C+c\cos B$$

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Symbols:

$\cos z$: cosine function, $a$: length, $b$: base, $c$: length, $B$: angle and $C$: angle

A&S Ref:

4.3.148

Permalink:

http://dlmf.nist.gov/4.42.E6

Encodings:

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See also:

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

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

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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:

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See also:

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

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

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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:

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See also:

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

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

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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

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