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21: 4.43 Cubic Equations

§4.43 Cubic Equations

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4.43.2 $$z^3+pz+q=0$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $p$: real constant and $q$: real constant

Referenced by:

(4.43.1), §4.43

Permalink:

http://dlmf.nist.gov/4.43.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants $C$ and $D$ given in Eq. (4.43.1), they read

1. $A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=C$, when and $C\le 1$.

2. $A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=C$, when and $C>1$.

3. $B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=D$, when $p>0$.

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for §4.43 and Ch.4

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$A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=4q/A^3$, when $4p^3+27q^2\le 0$.

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$A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=-4q/A^3$, when , , and $4p^3+27q^2>0$.

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$B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=-4q/B^3$, when $p>0$.

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22: 4.37 Inverse Hyperbolic Functions

… ► $\mathrm{Arcsinh}z$ and $\mathrm{Arccsch}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. … ►The principal values (or principal branches) of the inverse $\sinh$, $\cosh$, and $\tanh$ are obtained by introducing cuts in the $z$-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. …The principal branches are denoted by $\mathrm{arcsinh}$, $\mathrm{arccosh}$, $\mathrm{arctanh}$ respectively. 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. … ►For the corresponding results for $\mathrm{arccsch}z$, $\mathrm{arcsech}z$, and $\mathrm{arccoth}z$, use (4.37.7)–(4.37.9); compare §4.23(iv). …

23: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

► ►►

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

►

►

4.17.1 $$\underset{z\to 0}{lim}\frac{\sin z}{z}=1,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.74

Permalink:

http://dlmf.nist.gov/4.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

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4.17.2 $$\underset{z\to 0}{lim}\frac{\tan z}{z}=1.$$

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Symbols:

$\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.75

Permalink:

http://dlmf.nist.gov/4.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

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24: 22.10 Maclaurin Series

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22.10.4 $$\mathrm{sn}(z,k)=\sin z-\frac{k^2}{4}(z-\sin z\cos z)\cos z+O\left(k^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.1

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.5 $$\mathrm{cn}(z,k)=\cos z+\frac{k^2}{4}(z-\sin z\cos z)\sin z+O\left(k^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.2

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.7 $$\mathrm{sn}(z,k)=\tanh z-\frac{k_{}^{\prime }{}_{}{}^2}{4}(z-\sinh z\cosh z){\mathrm{sech}}^{2}z+O\left(k_{}^{\prime }{}_{}{}^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.1

Referenced by:

§22.10(ii)

Permalink:

http://dlmf.nist.gov/22.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.8 $$\mathrm{cn}(z,k)=\mathrm{sech}z+\frac{k_{}^{\prime }{}_{}{}^2}{4}(z-\sinh z\cosh z)\tanh z\mathrm{sech}z+O\left(k_{}^{\prime }{}_{}{}^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.2

Permalink:

http://dlmf.nist.gov/22.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.9 $$\mathrm{dn}(z,k)=\mathrm{sech}z+\frac{k_{}^{\prime }{}_{}{}^2}{4}(z+\sinh z\cosh z)\tanh z\mathrm{sech}z+O\left(k_{}^{\prime }{}_{}{}^4\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.3

Referenced by:

§22.10(ii)

Permalink:

http://dlmf.nist.gov/22.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

…

25: 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). … ► $\mathrm{Si}\left(z\right)$ is an odd entire function. …$\mathrm{Cin}\left(z\right)$ is an even entire function. … ►

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

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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

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6.2.18 $$\mathrm{g}\left(z\right)=-\mathrm{Ci}\left(z\right)\cos z-\mathrm{si}\left(z\right)\sin z.$$

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Defines:

$\mathrm{g}\left(z\right)$: 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:

5.2.7

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.2.E18

Encodings:

pMML, png, TeX

See also:

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

…

26: 4.22 Infinite Products and Partial Fractions

§4.22 Infinite Products and Partial Fractions

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4.22.1 $$\sin z=z\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z^2}{n^2{\pi }^2}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.89

Permalink:

http://dlmf.nist.gov/4.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

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4.22.2 $$\cos z=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{4z^2}{{(2n-1)}^2{\pi }^2}\right).$$

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Symbols:

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

A&S Ref:

4.3.90

Permalink:

http://dlmf.nist.gov/4.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

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4.22.3 $$\cot z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{1}{z^2-n^2{\pi }^2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.91

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/4.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

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4.22.4 $${\csc }^{2}z=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{{(z-n\pi )}^2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.92

Permalink:

http://dlmf.nist.gov/4.22.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

…

27: 4.24 Inverse Trigonometric Functions: Further Properties

§4.24 Inverse Trigonometric Functions: Further Properties

►

§4.24(i) Power Series

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§4.24(ii) Derivatives

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§4.24(iii) Addition Formulas

… ►

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

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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

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28: 4.26 Integrals

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§4.26(ii) Indefinite Integrals

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§4.26(iii) Definite Integrals

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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).

29: 19.11 Addition Theorems

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$\sin \psi ={\displaystyle \frac{(\sin \theta \cos \varphi )\mathrm{\Delta }(\varphi )+(\sin \varphi \cos \theta )\mathrm{\Delta }(\theta )}{1-k^2{\sin }^2\theta {\sin }^2\varphi }},$

… ►

$\cos \psi ={\displaystyle \frac{\cos \theta \cos \varphi -(\sin \theta \sin \varphi )\mathrm{\Delta }(\theta )\mathrm{\Delta }(\varphi )}{1-k^2{\sin }^2\theta {\sin }^2\varphi }},$

… ►In the case of $\theta ,\varphi \in [0,\pi /2)$ and , we can use … ►

$$\cos \psi =(\cos \left(2\theta \right)+k^2{\sin }^4\theta )/(1-k^2{\sin }^4\theta ),$$

… ►

$$\tan \theta =\tan \left(\frac{1}{2}\psi \right)\sqrt{\frac{1+\cos \psi }{(\cos \psi )+\mathrm{\Delta }(\psi )}},$$

…

30: 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

… ►For example, $\arcsin x=\arctan \left(x{(1-x^2)}^{-1/2}\right)$. … ►For $\mathrm{arccsch}$, $\mathrm{arcsech}$, and $\mathrm{arccoth}$ we have (4.37.7)–(4.37.9). …