inverse trigonometric functions

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21: 19.11 Addition Theorems

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19.11.6_5 $$R_C(\gamma -\delta ,\gamma )=\frac{-1}{\sqrt{\delta }}\arctan \left(\frac{\sqrt{\delta }\sin \theta \sin \varphi \sin \psi }{{\alpha }^2-1-{\alpha }^2\cos \theta \cos \varphi \cos \psi }\right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $\arctan z$: arctangent function, $\sin z$: sine function, $\varphi$: real or complex argument, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\psi$: angle, $\gamma$ and $\delta$

Referenced by:

§19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)

Permalink:

http://dlmf.nist.gov/19.11.E6_5

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

This equation was given a number.

See also:

Annotations for §19.11(i), §19.11 and Ch.19

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22: 14.15 Uniform Asymptotic Approximations

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14.15.21 $${\left(y-{\alpha }^2\right)}^{1/2}-\alpha \arctan \left(\frac{{\left(y-{\alpha }^2\right)}^{1/2}}{\alpha }\right)=\arccos \left(\frac{x}{{\left(1-{\alpha }^2\right)}^{1/2}}\right)-\frac{\alpha }{2}\arccos \left(\frac{\left(1+{\alpha }^2\right)x^2-1+{\alpha }^2}{\left(1-{\alpha }^2\right)\left(1-x^2\right)}\right),$$ $x\le {\left(1-{\alpha }^2\right)}^{1/2}$, $y\ge {\alpha }^2$,

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Symbols:

$\arccos z$: arccosine function, $\arctan z$: arctangent function, $x$: real variable, $y$ and $\alpha$

Referenced by:

§14.15(iv)

Permalink:

http://dlmf.nist.gov/14.15.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(iv), §14.15 and Ch.14

… ►where the inverse trigonometric functions take their principal values (§4.23(ii)). … ►

14.15.27 $$\frac{1}{2}\zeta {\left({\zeta }^2-{\alpha }^2\right)}^{1/2}-\frac{1}{2}{\alpha }^2\mathrm{arccosh}\left(\frac{\zeta }{\alpha }\right)={\left(1-a^2\right)}^{1/2}\mathrm{arctanh}\left(\frac{1}{x}{\left(\frac{x^2-a^2}{1-a^2}\right)}^{1/2}\right)-\mathrm{arccosh}\left(\frac{x}{a}\right),$$ , ,

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $x$: real variable, $a$ , $\zeta$ and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

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14.15.28 $$\frac{1}{2}{\alpha }^2\arcsin \left(\frac{\zeta }{\alpha }\right)+\frac{1}{2}\zeta {\left({\alpha }^2-{\zeta }^2\right)}^{1/2}=\arcsin \left(\frac{x}{a}\right)-{\left(1-a^2\right)}^{1/2}\arctan \left(x{\left(\frac{1-a^2}{a^2-x^2}\right)}^{1/2}\right),$$ $-a\le x\le a$, $-\alpha \le \zeta \le \alpha$,

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Symbols:

$\arcsin z$: arcsine function, $\arctan z$: arctangent function, $x$: real variable, $a$ , $\zeta$ and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

… ►The inverse hyperbolic and trigonometric functions take their principal values (§§4.23(ii), 4.37(ii)). …

23: 19.30 Lengths of Plane Curves

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19.30.12 $$s=aF(\varphi ,1/\sqrt{2}),$$ $\varphi =\arcsin \sqrt{2/(q+1)}=\arccos \left(\tan \theta \right)$.

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\arccos z$: arccosine function, $\arcsin z$: arcsine function, $\tan z$: tangent function, $\varphi$: real or complex argument, $q$ and $s$: arclength

Permalink:

http://dlmf.nist.gov/19.30.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.30(iii), §19.30 and Ch.19

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24: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

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13.21.12 $$\sqrt{\kappa \zeta -4{\mu }^2}-2\mu \arctan \left(\frac{\sqrt{\kappa \zeta -4{\mu }^2}}{2\mu }\right)=\frac{1}{2}(X-\pi \mu )-\mu \arctan \left(\frac{x\kappa -2{\mu }^2}{\mu X}\right)+\kappa \arcsin \left(\frac{X}{2\sqrt{{\kappa }^2-{\mu }^2}}\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arcsin z$: arcsine function, $\arctan z$: arctangent function, $x$: real variable and $X$

Permalink:

http://dlmf.nist.gov/13.21.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(ii), §13.21 and Ch.13

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13.21.20 $$\widehat{\zeta }=-{\left(\frac{3}{2\kappa }\left(-\frac{1}{2}X+2\mu \arctan \left(\frac{x\kappa -x\sqrt{{\kappa }^2-{\mu }^2}-2{\mu }^2}{\mu X}\right)+\kappa \arccos \left(\frac{x-2\kappa }{2\sqrt{{\kappa }^2-{\mu }^2}}\right)\right)\right)}^{2/3},$$ ,

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Symbols:

$\arccos z$: arccosine function, $\arctan z$: arctangent function, $x$: real variable and $X$

Permalink:

http://dlmf.nist.gov/13.21.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(iii), §13.21 and Ch.13

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25: 22.14 Integrals

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22.14.2 $$\int \mathrm{cn}(x,k)dx=k^{-1}\mathrm{Arccos}\left(\mathrm{dn}(x,k)\right),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arccos}z$: general arccosine function, $x$: real and $k$: modulus

A&S Ref:

16.24.2

Permalink:

http://dlmf.nist.gov/22.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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22.14.3 $$\int \mathrm{dn}(x,k)dx=\mathrm{Arcsin}\left(\mathrm{sn}(x,k)\right)=\mathrm{am}(x,k).$$

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Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real and $k$: modulus

A&S Ref:

16.24.3

Permalink:

http://dlmf.nist.gov/22.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

►The branches of the inverse trigonometric functions are chosen so that they are continuous. … ►

22.14.5 $$\int \mathrm{sd}(x,k)dx={(k{k}^{\prime })}^{-1}\mathrm{Arcsin}\left(-k\mathrm{cd}(x,k)\right),$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.24.5

Permalink:

http://dlmf.nist.gov/22.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

… ►Again, the branches of the inverse trigonometric functions must be continuous. …

26: 15.4 Special Cases

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15.4.3 $$F(\frac{1}{2},1;\frac{3}{2};-z^2)=z^{-1}\arctan z,$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\arctan z$: arctangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/15.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(i), §15.4 and Ch.15

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15.4.4 $$F(\frac{1}{2},\frac{1}{2};\frac{3}{2};z^2)=z^{-1}\arcsin z,$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\arcsin z$: arcsine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/15.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(i), §15.4 and Ch.15

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27: 4.39 Continued Fractions

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4.39.2 $$\frac{\mathrm{arcsinh}z}{\sqrt{1+z^2}}=\frac{z}{1+}\frac{1\cdot 2z^2}{3+}\frac{1\cdot 2z^2}{5+}\frac{3\cdot 4z^2}{7+}\frac{3\cdot 4z^2}{9+}\mathrm{\cdots },$$

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Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function and $z$: complex variable

A&S Ref:

4.6.36

Permalink:

http://dlmf.nist.gov/4.39.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.39 and Ch.4

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4.39.3 $$\mathrm{arctanh}z=\frac{z}{1-}\frac{z^2}{3-}\frac{4z^2}{5-}\frac{9z^2}{7-}\mathrm{\cdots },$$

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Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.6.35

Permalink:

http://dlmf.nist.gov/4.39.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.39 and Ch.4

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28: 13.20 Uniform Asymptotic Approximations for Large $\mu$

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13.20.9 $$\zeta \sqrt{{\zeta }^2+{\alpha }^2}+{\alpha }^2\mathrm{arcsinh}\left(\frac{\zeta }{\alpha }\right)=\frac{X}{\mu }-\frac{2\kappa }{\mu }\ln \left(\frac{X+x-2\kappa }{2\sqrt{{\mu }^2-{\kappa }^2}}\right)-2\ln \left(\frac{\mu X+2{\mu }^2-\kappa x}{x\sqrt{{\mu }^2-{\kappa }^2}}\right).$$

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Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iii), §13.20 and Ch.13

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13.20.13 $$\zeta \sqrt{{\zeta }^2-{\alpha }^2}-{\alpha }^2\mathrm{arccosh}\left(\frac{\zeta }{\alpha }\right)=\frac{X}{\mu }-\frac{2\kappa }{\mu }\ln \left(\frac{X+x-2\kappa }{2\sqrt{{\kappa }^2-{\mu }^2}}\right)-2\ln \left(\frac{\kappa x-\mu X-2{\mu }^2}{x\sqrt{{\kappa }^2-{\mu }^2}}\right),$$ $x\ge 2\kappa +2\sqrt{{\kappa }^2-{\mu }^2}$,

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

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13.20.14 $$\zeta \sqrt{{\alpha }^2-{\zeta }^2}+{\alpha }^2\arcsin \left(\frac{\zeta }{\alpha }\right)=\frac{X}{\mu }+\frac{2\kappa }{\mu }\arctan \left(\frac{x-2\kappa }{X}\right)-2\arctan \left(\frac{\kappa x-2{\mu }^2}{\mu X}\right),$$ $2\kappa -2\sqrt{{\kappa }^2-{\mu }^2}\le x\le 2\kappa +2\sqrt{{\kappa }^2-{\mu }^2}$,

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Symbols:

$\arcsin z$: arcsine function, $\arctan z$: arctangent function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

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13.20.15 $$-\zeta \sqrt{{\zeta }^2-{\alpha }^2}-{\alpha }^2\mathrm{arccosh}\left(-\frac{\zeta }{\alpha }\right)=-\frac{X}{\mu }+\frac{2\kappa }{\mu }\ln \left(\frac{2\kappa -X-x}{2\sqrt{{\kappa }^2-{\mu }^2}}\right)+2\ln \left(\frac{\mu X+2{\mu }^2-\kappa x}{x\sqrt{{\kappa }^2-{\mu }^2}}\right),$$ ,

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

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29: Guide to Searching the DLMF

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All the inverse trigonometric functions (arcsin vs. Arcsin, etc.).

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30: 10.68 Modulus and Phase Functions

§10.68 Modulus and Phase Functions

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§10.68(i) Definitions

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§10.68(ii) Basic Properties

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§10.68(iii) Asymptotic Expansions for Large Argument

… ►Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). …