cosine function

SearchAdvancedHelp

(0.013 seconds)

11—20 of 282 matching pages

11: 4.31 Special Values and Limits

… ►

4.31.3 $$\underset{z\to 0}{lim}\frac{\cosh z-1}{z^2}=\frac{1}{2}.$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/4.31.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4

12: 20.5 Infinite Products and Related Results

… ►

20.5.1 $${\theta }_1(z,q)=2q^{1/4}\sin z\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1-2q^{2n}\cos \left(2z\right)+q^{4n}\right),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome

Referenced by:

§20.4(i), §20.5(i), §23.17(iii)

Permalink:

http://dlmf.nist.gov/20.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

►

20.5.2 $${\theta }_2(z,q)=2q^{1/4}\cos z\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1+2q^{2n}\cos \left(2z\right)+q^{4n}\right),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

►

20.5.3 $${\theta }_3(z,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1+2q^{2n-1}\cos \left(2z\right)+q^{4n-2}\right),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex and $q$: nome

Referenced by:

§23.15(ii), §23.17(iii), §23.19

Permalink:

http://dlmf.nist.gov/20.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

►

20.5.4 $${\theta }_4(z,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1-2q^{2n-1}\cos \left(2z\right)+q^{4n-2}\right).$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex and $q$: nome

Referenced by:

§20.4(i), §20.5(i)

Permalink:

http://dlmf.nist.gov/20.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

… ►

20.5.10 $$\frac{{\theta }_1^{\prime }(z,q)}{{\theta }_1(z,q)}-\cot z=4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1-2q^{2n}\cos \left(2z\right)+q^{4n}}=4\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1-q^{2n}}\sin \left(2nz\right),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\cot z$: cotangent function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome

A&S Ref:

16.29.1

Referenced by:

§20.5(ii), §20.5(ii)

Permalink:

http://dlmf.nist.gov/20.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(ii), §20.5 and Ch.20

…

13: 28.32 Mathematical Applications

… ►

28.32.3 $$\frac{{\partial }^{2}V}{{\partial \xi }^2}+\frac{{\partial }^{2}V}{{\partial \eta }^2}+\frac{1}{2}{c}^2{k}^2(\cosh \left(2\xi \right)-\cos \left(2\eta \right))V=0.$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\eta$: variable, $\xi$: variable, $k$: parameter and $V(\xi ,\eta )$: solution

Referenced by:

§28.32(i), §28.33(ii)

Permalink:

http://dlmf.nist.gov/28.32.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

… ►

28.32.4 $$\frac{{\partial }^{2}K}{{\partial z}^2}-\frac{{\partial }^{2}K}{{\partial \zeta }^2}=2q\left(\cos \left(2z\right)-\cos \left(2\zeta \right)\right)K.$$

ⓘ

Symbols:

$\cos z$: cosine function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $q=h^2$: parameter, $z$: complex variable, $\zeta$: variable and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/28.32.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

…

14: 6.2 Definitions and Interrelations

… ►This is also true of the functions $\mathrm{Ci}\left(z\right)$ and $\mathrm{Chi}\left(z\right)$ defined in §6.2(ii). … ►

§6.2(iii) Auxiliary Functions

►

6.2.17 $$\mathrm{f}\left(z\right)=\mathrm{Ci}\left(z\right)\sin z-\mathrm{si}\left(z\right)\cos z,$$

ⓘ

Defines:

$\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals

Symbols:

$\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{si}\left(z\right)$: sine integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.6

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.2.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

… ►

6.2.19 $$\mathrm{Si}\left(z\right)=\frac{1}{2}\pi -\mathrm{f}\left(z\right)\cos z-\mathrm{g}\left(z\right)\sin z,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{Si}\left(z\right)$: sine integral and $z$: complex variable

A&S Ref:

5.2.8

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

►

6.2.20 $$\mathrm{Ci}\left(z\right)=\mathrm{f}\left(z\right)\sin z-\mathrm{g}\left(z\right)\cos z.$$

ⓘ

Symbols:

$\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

A&S Ref:

5.2.9

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.2.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

…

15: 19.10 Relations to Other Functions

… ►

19.10.2 $$(\sinh \varphi )R_C(1,{\cosh }^2\varphi )=\mathrm{gd}\left(\varphi \right).$$

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{gd}x$: Gudermannian function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.6(ii)

Permalink:

http://dlmf.nist.gov/19.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.10(ii), §19.10 and Ch.19

16: 4.37 Inverse Hyperbolic Functions

… ►

4.37.5 $$\mathrm{Arcsech}z=\mathrm{Arccosh}\left(1/z\right),$$

ⓘ

Defines:

$\mathrm{Arcsech}z$: general inverse hyperbolic secant function

Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(i), §4.37 and Ch.4

… ►Each is two-valued on the corresponding cut(s), and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts. … ►

4.37.8 $$\mathrm{arcsech}z=\mathrm{arccosh}\left(1/z\right).$$

ⓘ

Defines:

$\mathrm{arcsech}z$: inverse hyperbolic secant function

Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.6.5

Permalink:

http://dlmf.nist.gov/4.37.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(ii), §4.37 and Ch.4

… ►

4.37.27 $$z=\cosh w,$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/4.37.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(v), §4.37 and Ch.4

… ►

4.37.30 $$w=\mathrm{Arccosh}z=\pm \mathrm{arccosh}z+2k\pi \mathrm{i},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\mathrm{Arccosh}z$: general inverse hyperbolic cosine function, $k$: integer and $z$: complex variable

A&S Ref:

4.6.9

Permalink:

http://dlmf.nist.gov/4.37.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(v), §4.37 and Ch.4

…

17: 4.16 Elementary Properties

§4.16 Elementary Properties

… ►

Table 4.16.1: Signs of the trigonometric functions in the four quadrants.

► ►►

Quadrant	$\sin \theta ,\csc \theta$	$\cos \theta ,\sec \theta$	$\tan \theta ,\cot \theta$
…

►

►

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

► ►►

$x$	$-\theta$	$\frac{1}{2}\pi \pm \theta$	$\pi \pm \theta$	$\frac{3}{2}\pi \pm \theta$	$2\pi \pm \theta$
…

►

►

Table 4.16.3: Trigonometric functions: interrelations. …

► ►►

	$\sin \theta =a$	$\cos \theta =a$	$\tan \theta =a$	$\csc \theta =a$	$\sec \theta =a$	$\cot \theta =a$
…

►

18: 10.12 Generating Function and Associated Series

§10.12 Generating Function and Associated Series

… ►

$\cos \left(z\sin \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

… ►

$\cos \left(z\cos \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{(-1)}^k{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

►

$\sin \left(z\cos \theta \right)=2{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{J}_{2k+1}\left(z\right)\cos \left((2k+1)\theta \right).$

… ►

$\cos z=J_0\left(z\right)-2J_2\left(z\right)+2J_4\left(z\right)-2J_6\left(z\right)+\mathrm{\cdots },$

…

19: 34.8 Approximations for Large Parameters

… ►

34.8.1 $j_1,j_2,j_3\gg l_3\gg 1$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $o\left(x\right)$: order less than, $\sin z$: sine function, : $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $l,l_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §34.8 and Ch.34

… ►

34.8.2 $$\cos \theta =\frac{j_1(j_1+1)+j_2(j_2+1)-j_3(j_3+1)}{2\sqrt{j_1(j_1+1)j_2(j_2+1)}},$$

ⓘ

Symbols:

$\cos z$: cosine function and $j,j_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §34.8 and Ch.34

…

20: 4.20 Derivatives and Differential Equations

… ►

4.20.1 $$\frac{d}{dz}\sin z=\cos z,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.105

Permalink:

http://dlmf.nist.gov/4.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

►

4.20.2 $$\frac{d}{dz}\cos z=-\sin z,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.106

Permalink:

http://dlmf.nist.gov/4.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

… ►

4.20.8 $$\frac{d^n}{{dz}^n}\cos z=\cos \left(z+\frac{1}{2}n\pi \right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer and $z$: complex variable

A&S Ref:

4.3.112

Permalink:

http://dlmf.nist.gov/4.20.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

… ►

4.20.12 $$w=A\cos \left(az\right)+B\sin \left(az\right),$$

ⓘ

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.20.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

…