4 Elementary Functions Hyperbolic Functions 4.36 Infinite Products and Partial Fractions 4.38 Inverse Hyperbolic Functions: Further Properties

§4.37 Inverse Hyperbolic Functions

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Keywords:: inverse hyperbolic functions
Referenced by:: §4.2(i)
Permalink:: http://dlmf.nist.gov/4.37
See also:: Annotations for Ch.4

§4.37(i) General Definitions
§4.37(ii) Principal Values
§4.37(iii) Reflection Formulas
§4.37(iv) Logarithmic Forms
§4.37(v) Fundamental Property
§4.37(vi) Interrelations

§4.37(i) General Definitions

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Keywords:: analytic properties, branch points, definitions, general values, inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.37.i
See also:: Annotations for §4.37 and Ch.4

The general values of the inverse hyperbolic functions are defined by

4.37.1	$\displaystyle\operatorname{Arcsinh}z$	$\displaystyle=\int_{0}^{z}\frac{\,\mathrm{d}t}{(1+t^{2})^{1/2}},$
ⓘ Defines: $\operatorname{Arcsinh}\NVar{z}$ : general inverse hyperbolic sine function Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $z$ : complex variable Referenced by: §4.37(i), §4.37(ii) Permalink: http://dlmf.nist.gov/4.37.E1 Encodings: TeX, pMML, png See also: Annotations for §4.37(i), §4.37 and Ch.4
4.37.2	$\displaystyle\operatorname{Arccosh}z$	$\displaystyle=\int_{1}^{z}\frac{\,\mathrm{d}t}{(t^{2}-1)^{1/2}},$
ⓘ Defines: $\operatorname{Arccosh}\NVar{z}$ : general inverse hyperbolic cosine function Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $z$ : complex variable Referenced by: §4.37(i) Permalink: http://dlmf.nist.gov/4.37.E2 Encodings: TeX, pMML, png See also: Annotations for §4.37(i), §4.37 and Ch.4
4.37.3	$\displaystyle\operatorname{Arctanh}z$	$\displaystyle=\int_{0}^{z}\frac{\,\mathrm{d}t}{1-t^{2}},$
$z\neq\pm 1$ ,
ⓘ Defines: $\operatorname{Arctanh}\NVar{z}$ : general inverse hyperbolic tangent function Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $z$ : complex variable Referenced by: §4.37(i), §4.37(ii) Permalink: http://dlmf.nist.gov/4.37.E3 Encodings: TeX, pMML, png See also: Annotations for §4.37(i), §4.37 and Ch.4
4.37.4	$\displaystyle\operatorname{Arccsch}z$	$\displaystyle=\operatorname{Arcsinh}\left(1/z\right),$
ⓘ Defines: $\operatorname{Arccsch}\NVar{z}$ : general inverse hyperbolic cosecant function Symbols: $\operatorname{Arcsinh}\NVar{z}$ : general inverse hyperbolic sine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.37.E4 Encodings: TeX, pMML, png See also: Annotations for §4.37(i), §4.37 and Ch.4
4.37.5	$\displaystyle\operatorname{Arcsech}z$	$\displaystyle=\operatorname{Arccosh}\left(1/z\right),$
ⓘ Defines: $\operatorname{Arcsech}\NVar{z}$ : general inverse hyperbolic secant function Symbols: $\operatorname{Arccosh}\NVar{z}$ : general inverse hyperbolic cosine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.37.E5 Encodings: TeX, pMML, png See also: Annotations for §4.37(i), §4.37 and Ch.4
4.37.6	$\displaystyle\operatorname{Arccoth}z$	$\displaystyle=\operatorname{Arctanh}\left(1/z\right).$
ⓘ Defines: $\operatorname{Arccoth}\NVar{z}$ : general inverse hyperbolic cotangent function Symbols: $\operatorname{Arctanh}\NVar{z}$ : general inverse hyperbolic tangent function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.37.E6 Encodings: TeX, pMML, png See also: Annotations for §4.37(i), §4.37 and Ch.4

In (4.37.1) the integration path may not pass through either of the points $t=\pm i$ , and the function $(1+t^{2})^{1/2}$ assumes its principal value when $t$ is real. In (4.37.2) the integration path may not pass through either of the points $\pm 1$ , and the function $(t^{2}-1)^{1/2}$ assumes its principal value when $t\in(1,\infty)$ . Elsewhere on the integration paths in (4.37.1) and (4.37.2) the branches are determined by continuity. In (4.37.3) the integration path may not intersect $\pm 1$ . Each of the six functions is a multivalued function of $z$ . $\operatorname{Arcsinh}z$ and $\operatorname{Arccsch}z$ have branch points at $z=\pm i$ ; the other four functions have branch points at $z=\pm 1$ .

§4.37(ii) Principal Values

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Defines:: $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function and $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function
Keywords:: definitions, inverse hyperbolic functions, principal values
Notes:: See Levinson and Redheffer (1970, pp. 68–69).
Referenced by:: §14.15(v), §19.2(iv)
Permalink:: http://dlmf.nist.gov/4.37.ii
See also:: Annotations for §4.37 and Ch.4

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. Compare the principal value of the logarithm (§4.2(i)). The principal branches are denoted by $\operatorname{arcsinh}$ , $\operatorname{arccosh}$ , $\operatorname{arctanh}$ 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.

The principal values of the inverse hyperbolic cosecant, hyperbolic secant, and hyperbolic tangent are given by

4.37.7	$\displaystyle\operatorname{arccsch}z$	$\displaystyle=\operatorname{arcsinh}\left(1/z\right),$
ⓘ Defines: $\operatorname{arccsch}\NVar{z}$ : inverse hyperbolic cosecant function Symbols: $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function and $z$ : complex variable A&S Ref: 4.6.4 Referenced by: §4.37(iv), §4.45(i), §4.45(ii) Permalink: http://dlmf.nist.gov/4.37.E7 Encodings: TeX, pMML, png See also: Annotations for §4.37(ii), §4.37 and Ch.4
4.37.8	$\displaystyle\operatorname{arcsech}z$	$\displaystyle=\operatorname{arccosh}\left(1/z\right).$
ⓘ Defines: $\operatorname{arcsech}\NVar{z}$ : inverse hyperbolic secant function Symbols: $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function and $z$ : complex variable A&S Ref: 4.6.5 Permalink: http://dlmf.nist.gov/4.37.E8 Encodings: TeX, pMML, png See also: Annotations for §4.37(ii), §4.37 and Ch.4
4.37.9	$\displaystyle\operatorname{arccoth}z$	$\displaystyle=\operatorname{arctanh}\left(1/z\right),$
$z\neq\pm 1$ .
ⓘ Defines: $\operatorname{arccoth}\NVar{z}$ : inverse hyperbolic cotangent function Symbols: $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function and $z$ : complex variable A&S Ref: 4.6.6 Referenced by: §4.37(iv), §4.45(i), §4.45(ii) Permalink: http://dlmf.nist.gov/4.37.E9 Encodings: TeX, pMML, png See also: Annotations for §4.37(ii), §4.37 and Ch.4

These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively.

Except where indicated otherwise, it is assumed throughout the DLMF that the inverse hyperbolic functions assume their principal values.

See accompanying text — Figure 4.37.1: $z$ -plane. Branch cuts for the inverse hyperbolic functions. Magnify

Graphs of the principal values for real arguments are given in §4.29. This section also indicates conformal mappings, and surface plots for complex arguments.

§4.37(iii) Reflection Formulas

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Keywords:: inverse hyperbolic functions, reflection formulas
Notes:: (4.37.11) follows from (4.37.19).
Permalink:: http://dlmf.nist.gov/4.37.iii
See also:: Annotations for §4.37 and Ch.4

4.37.10	$\displaystyle\operatorname{arcsinh}\left(-z\right)$	$\displaystyle=-\operatorname{arcsinh}z.$
ⓘ Symbols: $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function and $z$ : complex variable A&S Ref: 4.6.11 Permalink: http://dlmf.nist.gov/4.37.E10 Encodings: TeX, pMML, png See also: Annotations for §4.37(iii), §4.37 and Ch.4
4.37.11	$\displaystyle\operatorname{arccosh}\left(-z\right)$	$\displaystyle=\pm\pi i+\operatorname{arccosh}z,$
$\Im z\gtrless 0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit and $z$ : complex variable A&S Ref: 4.6.12 (has an error even in the tenth printing.) Referenced by: §4.37(iii) Permalink: http://dlmf.nist.gov/4.37.E11 Encodings: TeX, pMML, png See also: Annotations for §4.37(iii), §4.37 and Ch.4
4.37.12	$\displaystyle\operatorname{arctanh}\left(-z\right)$	$\displaystyle=-\operatorname{arctanh}z,$
$z\neq\pm 1$ .
ⓘ Symbols: $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function and $z$ : complex variable A&S Ref: 4.6.13 Permalink: http://dlmf.nist.gov/4.37.E12 Encodings: TeX, pMML, png See also: Annotations for §4.37(iii), §4.37 and Ch.4
4.37.13	$\displaystyle\operatorname{arccsch}\left(-z\right)$	$\displaystyle=-\operatorname{arccsch}z.$
ⓘ Symbols: $\operatorname{arccsch}\NVar{z}$ : inverse hyperbolic cosecant function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.37.E13 Encodings: TeX, pMML, png See also: Annotations for §4.37(iii), §4.37 and Ch.4
4.37.14	$\displaystyle\operatorname{arcsech}\left(-z\right)$	$\displaystyle=\mp\pi i+\operatorname{arcsech}z,$
$\Im z\gtrless 0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arcsech}\NVar{z}$ : inverse hyperbolic secant function, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.37.E14 Encodings: TeX, pMML, png See also: Annotations for §4.37(iii), §4.37 and Ch.4
4.37.15	$\displaystyle\operatorname{arccoth}\left(-z\right)$	$\displaystyle=-\operatorname{arccoth}z,$
$z\neq\pm 1$ .
ⓘ Symbols: $\operatorname{arccoth}\NVar{z}$ : inverse hyperbolic cotangent function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.37.E15 Encodings: TeX, pMML, png See also: Annotations for §4.37(iii), §4.37 and Ch.4

§4.37(iv) Logarithmic Forms

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Keywords:: inverse hyperbolic functions, logarithmic forms, values on the cuts
Notes:: The equations in this subsection may be verified in a similar manner to those of §4.23(iv). The only new feature is that in (4.37.19) the principal value of $(z^{2}-1)^{1/2}$ is discontinuous on the imaginary axis, hence to continue $(z^{2}-1)^{1/2}$ analytically we switch to the other branch. This accounts for the $\pm$ sign in (4.37.19).
Referenced by:: §4.38(ii), §4.45(i), §4.45(ii)
Permalink:: http://dlmf.nist.gov/4.37.iv
See also:: Annotations for §4.37 and Ch.4

Throughout this subsection all quantities assume their principal values.

Inverse Hyperbolic Sine

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See also:: Annotations for §4.37(iv), §4.37 and Ch.4

4.37.16		$\operatorname{arcsinh}z=\ln\left((z^{2}+1)^{1/2}+z\right),$
$z/i\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$ ;
ⓘ Symbols: $\mathbb{C}$ : complex plane, $\in$ : element of, $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.37.E16 Encodings: TeX, pMML, png See also: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

compare Figure 4.37.1(i). On the cuts

4.37.17		$\operatorname{arcsinh}\left(iy\right)=\tfrac{1}{2}\pi i\pm\ln\left((y^{2}-1)^{% 1/2}+y\right),$
$y\in[1,\infty)$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b})$ : half-closed interval, $\in$ : element of, $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function and $y$ : real variable Permalink: http://dlmf.nist.gov/4.37.E17 Encodings: TeX, pMML, png See also: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

4.37.18		$\operatorname{arcsinh}\left(iy\right)=-\tfrac{1}{2}\pi i\pm\ln\left((y^{2}-1)^% {1/2}-y\right),$
$y\in(-\infty,-1]$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b}]$ : half-closed interval and $y$ : real variable Permalink: http://dlmf.nist.gov/4.37.E18 Encodings: TeX, pMML, png See also: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

the upper/lower signs corresponding to the right/left sides.

Inverse Hyperbolic Cosine

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See also:: Annotations for §4.37(iv), §4.37 and Ch.4

4.37.19		$\operatorname{arccosh}z=\ln\left(\pm(z^{2}-1)^{1/2}+z\right),$
$z\in\mathbb{C}\setminus(-\infty,1)$ ,
ⓘ Symbols: $\mathbb{C}$ : complex plane, $\in$ : element of, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction and $z$ : complex variable Referenced by: §4.37(iv), §4.37(iii), §4.37(iv) Permalink: http://dlmf.nist.gov/4.37.E19 Encodings: TeX, pMML, png See also: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

the upper or lower sign being taken according as $\Re z\gtrless 0$ ; compare Figure 4.37.1(ii). Also,

4.37.20		$\operatorname{arccosh}\left(\mathrm{i}y\right)=\pm\tfrac{1}{2}\pi\mathrm{i}+% \ln\left((y^{2}+1)^{1/2}\pm y\right),$
$y\gtrless 0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function and $y$ : real variable Referenced by: §4.37(iv) Permalink: http://dlmf.nist.gov/4.37.E20 Encodings: TeX, pMML, png See also: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on $(-\infty,1]$ . Compare Figure 4.37.1(ii).

An equivalent definition is

4.37.21		$\operatorname{arccosh}z=2\ln\left(\left(\frac{z+1}{2}\right)^{1/2}+\left(\frac% {z-1}{2}\right)^{1/2}\right),$
$z\in\mathbb{C}\setminus(-\infty,1)$ ;
ⓘ Symbols: $\mathbb{C}$ : complex plane, $\in$ : element of, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.37.E21 Encodings: TeX, pMML, png See also: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

see Kahan (1987).

On the part of the cuts from $-1$ to $1$

4.37.22		$\operatorname{arccosh}x=\pm\ln\left(i(1-x^{2})^{1/2}+x\right),$
$x\in(-1,1]$ ,
ⓘ Symbols: $\in$ : element of, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b}]$ : half-closed interval and $x$ : real variable Permalink: http://dlmf.nist.gov/4.37.E22 Encodings: TeX, pMML, png See also: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

the upper/lower sign corresponding to the upper/lower side.

On the part of the cut from $-\infty$ to $-1$

4.37.23		$\operatorname{arccosh}x=\pm\pi i+\ln\left((x^{2}-1)^{1/2}-x\right),$
$x\in(-\infty,-1]$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b}]$ : half-closed interval and $x$ : real variable Permalink: http://dlmf.nist.gov/4.37.E23 Encodings: TeX, pMML, png See also: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

the upper/lower sign corresponding to the upper/lower side.

Inverse Hyperbolic Tangent

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See also:: Annotations for §4.37(iv), §4.37 and Ch.4

4.37.24		$\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right),$
$z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$ ;
ⓘ Symbols: $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\in$ : element of, $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.37.E24 Encodings: TeX, pMML, png See also: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

compare Figure 4.37.1(iii). On the cuts

4.37.25		$\operatorname{arctanh}x=\pm\tfrac{1}{2}\pi i+\tfrac{1}{2}\ln\left(\frac{x+1}{x% -1}\right),$
$x\in(-\infty,-1)\cup(1,\infty)$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\cup$ : union and $x$ : real variable Permalink: http://dlmf.nist.gov/4.37.E25 Encodings: TeX, pMML, png See also: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

the upper/lower sign corresponding to the upper/lower sides.

Other Inverse Functions

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Keywords:: inverse hyperbolic functions, logarithmic forms, values on the cuts
See also:: Annotations for §4.37(iv), §4.37 and Ch.4

For the corresponding results for $\operatorname{arccsch}z$ , $\operatorname{arcsech}z$ , and $\operatorname{arccoth}z$ , use (4.37.7)–(4.37.9); compare §4.23(iv).

§4.37(v) Fundamental Property

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Keywords:: fundamental property, inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.37.v
See also:: Annotations for §4.37 and Ch.4

With $k\in\mathbb{Z}$ , the general solutions of the equations

4.37.26		$z=\sinh w,$
ⓘ Symbols: $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.37.E26 Encodings: TeX, pMML, png See also: Annotations for §4.37(v), §4.37 and Ch.4

4.37.27		$z=\cosh w,$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.37.E27 Encodings: TeX, pMML, png See also: Annotations for §4.37(v), §4.37 and Ch.4

4.37.28		$z=\tanh w,$
ⓘ Symbols: $\tanh\NVar{z}$ : hyperbolic tangent function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.37.E28 Encodings: TeX, pMML, png See also: Annotations for §4.37(v), §4.37 and Ch.4

are respectively given by

4.37.29	$\displaystyle w$	$\displaystyle=\operatorname{Arcsinh}z=(-1)^{k}\operatorname{arcsinh}z+k\pi i,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\operatorname{Arcsinh}\NVar{z}$ : general inverse hyperbolic sine function, $k$ : integer and $z$ : complex variable A&S Ref: 4.6.8 Permalink: http://dlmf.nist.gov/4.37.E29 Encodings: TeX, pMML, png See also: Annotations for §4.37(v), §4.37 and Ch.4
4.37.30	$\displaystyle w$	$\displaystyle=\operatorname{Arccosh}z=\pm\operatorname{arccosh}z+2k\pi i,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\operatorname{Arccosh}\NVar{z}$ : 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: TeX, pMML, png See also: Annotations for §4.37(v), §4.37 and Ch.4
4.37.31	$\displaystyle w$	$\displaystyle=\operatorname{Arctanh}z=\operatorname{arctanh}z+k\pi i,$
$z\neq\pm 1$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function, $\mathrm{i}$ : imaginary unit, $\operatorname{Arctanh}\NVar{z}$ : general inverse hyperbolic tangent function, $k$ : integer and $z$ : complex variable A&S Ref: 4.6.10 Permalink: http://dlmf.nist.gov/4.37.E31 Encodings: TeX, pMML, png See also: Annotations for §4.37(v), §4.37 and Ch.4

§4.37(vi) Interrelations

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Keywords:: interrelations, inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.37.vi
See also:: Annotations for §4.37 and Ch.4

Table 4.30.1 can also be used to find interrelations between inverse hyperbolic functions. For example, $\operatorname{arcsech}a=\operatorname{arccoth}\left((1-a^{2})^{-1/2}\right)$ .

4.36 Infinite Products and Partial Fractions 4.38 Inverse Hyperbolic Functions: Further Properties

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(i) $\operatorname{arcsinh}z$	(ii) $\operatorname{arccosh}z$	(iii) $\operatorname{arctanh}z$

(iv) $\operatorname{arccsch}z$	(v) $\operatorname{arcsech}z$	(vi) $\operatorname{arccoth}z$