trigonometric functions

SearchAdvancedHelp

(0.018 seconds)

21—30 of 339 matching pages

21: 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

22: 4.34 Derivatives and Differential Equations

… ►

4.34.1 $$\frac{d}{dz}\sinh z=\cosh z,$$

ⓘ

Symbols:

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

A&S Ref:

4.5.71

Permalink:

http://dlmf.nist.gov/4.34.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

►

4.34.2 $$\frac{d}{dz}\cosh z=\sinh z,$$

ⓘ

Symbols:

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

A&S Ref:

4.5.72

Permalink:

http://dlmf.nist.gov/4.34.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

►

4.34.3 $$\frac{d}{dz}\tanh z={\mathrm{sech}}^{2}z,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.73

Permalink:

http://dlmf.nist.gov/4.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

►

4.34.4 $$\frac{d}{dz}\mathrm{csch}z=-\mathrm{csch}z\coth z,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{csch}z$: hyperbolic cosecant function, $\coth z$: hyperbolic cotangent function and $z$: complex variable

A&S Ref:

4.5.74

Permalink:

http://dlmf.nist.gov/4.34.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

… ►

4.34.11 $$w=A\cosh \left(az\right)+B\sinh \left(az\right),$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $a$: real or complex constant, $z$: complex variable, $w$: solution, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

…

23: 4.25 Continued Fractions

§4.25 Continued Fractions

►

4.25.1 $$\tan z=\frac{z}{1-}\frac{z^2}{3-}\frac{z^2}{5-}\frac{z^2}{7-}\mathrm{\cdots },$$ $z\ne \pm \frac{1}{2}\pi$, $\pm \frac{3}{2}\pi$, $\mathrm{\dots }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.94

Permalink:

http://dlmf.nist.gov/4.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

… ►

4.25.3 $$\frac{\arcsin z}{\sqrt{1-z^2}}=\frac{z}{1-}\frac{1\cdot 2z^2}{3-}\frac{1\cdot 2z^2}{5-}\frac{3\cdot 4z^2}{7-}\frac{3\cdot 4z^2}{9-}\mathrm{\cdots },$$

ⓘ

Symbols:

$\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.44

Permalink:

http://dlmf.nist.gov/4.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

… ►

4.25.4 $$\arctan z=\frac{z}{1+}\frac{z^2}{3+}\frac{4z^2}{5+}\frac{9z^2}{7+}\frac{16z^2}{9+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\arctan z$: arctangent function and $z$: complex variable

A&S Ref:

4.4.43

Referenced by:

§3.10(ii)

Permalink:

http://dlmf.nist.gov/4.25.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

… ►See Lorentzen and Waadeland (1992, pp. 560–571) for other continued fractions involving inverse trigonometric functions. …

24: 4.45 Methods of Computation

… ►

Trigonometric Functions

… ►The other trigonometric functions can be found from the definitions (4.14.4)–(4.14.7). ►

Inverse Trigonometric Functions

►The function $\arctan x$ can always be computed from its ascending power series after preliminary transformations to reduce the size of $x$. … ►For the remaining inverse trigonometric functions, we may use the identities provided by the fourth row of Table 4.16.3. …

25: 4.30 Elementary Properties

§4.30 Elementary Properties

►

Table 4.30.1: Hyperbolic functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.

► ►►►

	$\sinh \theta =a$	$\cosh \theta =a$	$\tanh \theta =a$	$\mathrm{csch}\theta =a$	$\mathrm{sech}\theta =a$	$\coth \theta =a$
$\sinh \theta$	$a$	${(a^2-1)}^{1/2}$	$a{(1-a^2)}^{-1/2}$	$a^{-1}$	$a^{-1}{(1-a^2)}^{1/2}$	${(a^2-1)}^{-1/2}$
…

►

26: 4.26 Integrals

… ►

§4.26(ii) Indefinite Integrals

… ►

§4.26(iii) Definite Integrals

… ►

Orthogonality Properties

… ►

§4.26(iv) Inverse Trigonometric Functions

… ►Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2000, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).

27: 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

… ►

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

ⓘ

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

►

20.5.11 $$\frac{{\theta }_2^{\prime }(z,q)}{{\theta }_2(z,q)}+\tan z=-4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1+2q^{2n}\cos \left(2z\right)+q^{4n}}=4\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{q^{2n}}{1-q^{2n}}\sin \left(2nz\right).$$

ⓘ

Symbols:

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

A&S Ref:

16.29.2

Referenced by:

§20.5(ii)

Permalink:

http://dlmf.nist.gov/20.5.E11

Encodings:

pMML, png, TeX

See also:

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

… ►

20.5.12 $$\frac{{\theta }_3^{\prime }(z,q)}{{\theta }_3(z,q)}=-4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{1+2q^{2n-1}\cos \left(2z\right)+q^{4n-2}}=4\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{q^n}{1-q^{2n}}\sin \left(2nz\right),$$

ⓘ

Symbols:

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

A&S Ref:

16.29.3

Permalink:

http://dlmf.nist.gov/20.5.E12

Encodings:

pMML, png, TeX

See also:

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

►

20.5.13 $$\frac{{\theta }_4^{\prime }(z,q)}{{\theta }_4(z,q)}=4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{1-2q^{2n-1}\cos \left(2z\right)+q^{4n-2}}=4\sum _{n=1}^{\mathrm{\infty }}\frac{q^n}{1-q^{2n}}\sin \left(2nz\right).$$

ⓘ

Symbols:

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

A&S Ref:

16.29.4

Referenced by:

§20.5(ii)

Permalink:

http://dlmf.nist.gov/20.5.E13

Encodings:

pMML, png, TeX

See also:

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

…

28: 4.19 Maclaurin Series and Laurent Series

§4.19 Maclaurin Series and Laurent Series

►

4.19.1 $$\sin z=z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+\mathrm{\cdots },$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.65

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

►

4.19.2 $$\cos z=1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+\mathrm{\cdots }.$$

ⓘ

Symbols:

$\cos z$: cosine function, $!$: factorial (as in $n!$) and $z$: complex variable

A&S Ref:

4.3.66

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

… ►

4.19.7 $$\ln \left(\frac{\sin z}{z}\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{2}^{2n-1}{B}_{2n}}{n(2n)!}{z}^{2n},$$ ,

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.71

Permalink:

http://dlmf.nist.gov/4.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

… ►

4.19.9 $$\ln \left(\frac{\tan z}{z}\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}{2}^{2n}(2^{2n-1}-1)B_{2n}}{n(2n)!}{z}^{2n},$$ .

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\tan z$: tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.73

Referenced by:

§4.19, §4.33

Permalink:

http://dlmf.nist.gov/4.19.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

29: 4.24 Inverse Trigonometric Functions: Further Properties

§4.24 Inverse Trigonometric Functions: Further Properties

►

§4.24(i) Power Series

… ►

§4.24(ii) Derivatives

… ►

§4.24(iii) Addition Formulas

… ►

4.24.17 $$\mathrm{Arctan}u\pm \mathrm{Arccot}v=\mathrm{Arctan}\left(\frac{uv\pm 1}{v\mp u}\right)=\mathrm{Arccot}\left(\frac{v\mp u}{uv\pm 1}\right).$$

ⓘ

Symbols:

$\mathrm{Arccot}z$: general arccotangent function and $\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.36

Permalink:

http://dlmf.nist.gov/4.24.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

…

30: 10.12 Generating Function and Associated Series

§10.12 Generating Function and Associated Series

… ►

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

►

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

►

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

►

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

…