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21: 4.38 Inverse Hyperbolic Functions: Further Properties

§4.38 Inverse Hyperbolic Functions: Further Properties

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§4.38(i) Power Series

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§4.38(ii) Derivatives

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§4.38(iii) Addition Formulas

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4.38.19 $$\mathrm{Arctanh}u\pm \mathrm{Arccoth}v=\mathrm{Arctanh}\left(\frac{uv\pm 1}{v\pm u}\right)=\mathrm{Arccoth}\left(\frac{v\pm u}{uv\pm 1}\right).$$

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Symbols:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function and $\mathrm{Arctanh}z$: general inverse hyperbolic tangent function

A&S Ref:

4.6.30

Permalink:

http://dlmf.nist.gov/4.38.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

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22: 1.10 Functions of a Complex Variable

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§1.10(vii) Inverse Functions

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Lagrange Inversion Theorem

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1.10.13 $$F(w)=z_0+\sum _{n=1}^{\mathrm{\infty }}{F}_n{(w-w_0)}^n$$

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Defines:

$F(w)$: inverse function of $f(z)$ (locally)

Symbols:

$z$: variable, $w$: variable, $n$: nonnegative integer and $F_n$: residue

Referenced by:

§1.10(vii)

Permalink:

http://dlmf.nist.gov/1.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(vii), §1.10(vii), §1.10 and Ch.1

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1.10.14 $$g(F(w))=g(z_0)+\sum _{n=1}^{\mathrm{\infty }}{G}_n{(w-w_0)}^n,$$

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Symbols:

$z$: variable, $w$: variable, $n$: nonnegative integer, $F(w)$: inverse function of $f(z)$ and $G_n$: residue

Referenced by:

§1.10(vii)

Permalink:

http://dlmf.nist.gov/1.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(vii), §1.10(vii), §1.10 and Ch.1

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Extended Inversion Theorem

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23: 4.40 Integrals

§4.40 Integrals

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§4.40(iv) Inverse Hyperbolic Functions

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4.40.11 $$\int \mathrm{arcsinh}xdx=x\mathrm{arcsinh}x-{(1+x^2)}^{1/2}.$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.6.43 (modified)

Permalink:

http://dlmf.nist.gov/4.40.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iv), §4.40 and Ch.4

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4.40.14 $$\int \mathrm{arccsch}xdx=x\mathrm{arccsch}x+\mathrm{arcsinh}x,$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.6.46 (modified. The first printing had an error.)

Permalink:

http://dlmf.nist.gov/4.40.E14

Encodings:

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See also:

Annotations for §4.40(iv), §4.40 and Ch.4

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4.40.15 $$\int \mathrm{arcsech}xdx=x\mathrm{arcsech}x+\arcsin x,$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\int$: integral, $\arcsin z$: arcsine function and $x$: real variable

A&S Ref:

4.6.47 (modified. The first printing had an error.)

Permalink:

http://dlmf.nist.gov/4.40.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iv), §4.40 and Ch.4

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24: 19.6 Special Cases

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19.6.8 $$F(\varphi ,1)=(\sin \varphi )R_C(1,{\cos }^2\varphi )={\mathrm{gd}}^{-1}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.10(ii)

Permalink:

http://dlmf.nist.gov/19.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(ii), §19.6 and Ch.19

►For the inverse Gudermannian function ${\mathrm{gd}}^{-1}\left(\varphi \right)$ see §4.23(viii). …

25: 4.25 Continued Fractions

§4.25 Continued Fractions

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

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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

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

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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:

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See also:

Annotations for §4.25 and Ch.4

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4.25.5 $${\mathrm{e}}^{2a\arctan \left(1/z\right)}=1+\frac{2a}{z-a+}\frac{a^2+1}{3z+}\frac{a^2+4}{5z+}\frac{a^2+9}{7z+}\mathrm{\cdots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\arctan z$: arctangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.42

Permalink:

http://dlmf.nist.gov/4.25.E5

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. …

26: 22.20 Methods of Computation

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22.20.4 $${\varphi }_{n-1}=\frac{1}{2}\left({\varphi }_n+\arcsin \left(\frac{c_n}{a_n}\sin {\varphi }_n\right)\right),$$

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Symbols:

$\arcsin z$: arcsine function, $\sin z$: sine function, $a_n$: numbers, $c_n$: numbers, $n$: positive and ${\varphi }_N$: approximations

Referenced by:

§22.20(ii)

Permalink:

http://dlmf.nist.gov/22.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.20(ii), §22.20 and Ch.22

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§22.20(v) Inverse Functions

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27: 4.26 Integrals

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4.26.5 $$\int \sec xdx={\mathrm{gd}}^{-1}\left(x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sec z$: secant function and $x$: real variable

A&S Ref:

4.3.117

Permalink:

http://dlmf.nist.gov/4.26.E5

Encodings:

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See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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§4.26(iv) Inverse Trigonometric Functions

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4.26.14 $$\int \arcsin xdx=x\arcsin x+{(1-x^2)}^{1/2},$$ ,

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\arcsin z$: arcsine function and $x$: real variable

A&S Ref:

4.4.58 (modified)

Permalink:

http://dlmf.nist.gov/4.26.E14

Encodings:

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See also:

Annotations for §4.26(iv), §4.26 and Ch.4

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4.26.15 $$\int \arccos xdx=x\arccos x-{(1-x^2)}^{1/2},$$ .

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\arccos z$: arccosine function and $x$: real variable

A&S Ref:

4.4.59 (modified)

Permalink:

http://dlmf.nist.gov/4.26.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iv), §4.26 and Ch.4

… ►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).

28: 19.2 Definitions

… ►In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). When $x$ and $y$ are positive, $R_C(x,y)$ is an inverse circular function if and an inverse hyperbolic function (or logarithm) if $x>y$: ►

19.2.18 $$R_C(x,y)=\frac{1}{\sqrt{y-x}}\arctan \sqrt{\frac{y-x}{x}}=\frac{1}{\sqrt{y-x}}\arccos \sqrt{x/y},$$ ,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\arccos z$: arccosine function and $\arctan z$: arctangent function

Referenced by:

§19.2(iv)

Permalink:

http://dlmf.nist.gov/19.2.E18

Encodings:

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See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.19 $$R_C(x,y)=\frac{1}{\sqrt{x-y}}\mathrm{arctanh}\sqrt{\frac{x-y}{x}}=\frac{1}{\sqrt{x-y}}\ln \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}},$$ .

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $\ln z$: principal branch of logarithm function

Referenced by:

§19.26(i), Figure 19.3.2, Figure 19.3.2, §19.36(ii)

Permalink:

http://dlmf.nist.gov/19.2.E19

Encodings:

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See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.20 $$R_C(x,y)=\sqrt{\frac{x}{x-y}}{R}_C(x-y,-y)=\frac{1}{\sqrt{x-y}}\mathrm{arctanh}\sqrt{\frac{x}{x-y}}=\frac{1}{\sqrt{x-y}}\ln \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}},$$ .

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $\ln z$: principal branch of logarithm function

Referenced by:

§19.2(iv), §19.20(iii), §19.25(i), Figure 19.3.2, Figure 19.3.2, §19.36(i)

Permalink:

http://dlmf.nist.gov/19.2.E20

Encodings:

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See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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29: 19.11 Addition Theorems

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19.11.5 $$\mathrm{\Pi }(\theta ,{\alpha }^2,k)+\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\mathrm{\Pi }(\psi ,{\alpha }^2,k)-{\alpha }^2{R}_C(\gamma -\delta ,\gamma ),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\psi$: angle, $\gamma$ and $\delta$

Referenced by:

§19.11(i), §19.11(i), Erratum (V1.0.24) for Subsection 19.11(i)

Permalink:

http://dlmf.nist.gov/19.11.E5

Encodings:

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See also:

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

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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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19.11.10 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\mathrm{\Pi }({\alpha }^2,k)-\mathrm{\Pi }(\theta ,{\alpha }^2,k)-{\alpha }^2{R}_C(\gamma -\delta ,\gamma ),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\gamma$ and $\delta$

Permalink:

http://dlmf.nist.gov/19.11.E10

Encodings:

pMML, png, TeX

See also:

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

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19.11.16 $$\mathrm{\Pi }(\psi ,{\alpha }^2,k)=2\mathrm{\Pi }(\theta ,{\alpha }^2,k)+{\alpha }^2{R}_C(\gamma -\delta ,\gamma ),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\delta$ , $\psi$: angle and $\gamma$

Permalink:

http://dlmf.nist.gov/19.11.E16

Encodings:

pMML, png, TeX

See also:

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

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30: 28.26 Asymptotic Approximations for Large $q$

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28.26.3 $$\varphi =2h\sinh z-\left(m+\frac{1}{2}\right)\arctan \left(\sinh z\right).$$

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Symbols:

$\sinh z$: hyperbolic sine function, $\arctan z$: arctangent function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.8 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E3

Encodings:

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See also:

Annotations for §28.26(i), §28.26 and Ch.28

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