# §4.37 Inverse Hyperbolic Functions

## §4.37(i) General Definitions

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 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 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 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 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 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 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

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 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 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 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.

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

 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 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 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 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 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 4 4.37.14 $\displaystyle\operatorname{arcsech}\left(-z\right)$ $\displaystyle=\mp\pi i+\operatorname{arcsech}z,$ $\Im z\gtrless 0$. 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 4

## §4.37(iv) Logarithmic Forms

Throughout this subsection all quantities assume their principal values.

### Inverse Hyperbolic Sine

 4.37.16 $\operatorname{arcsinh}z=\ln\left((z^{2}+1)^{1/2}+z\right),$ $z/i\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$;

compare Figure 4.37.1(i). On the cuts

 4.37.17 $\operatorname{arcsinh}\left(iy\right)=\tfrac{1}{2}\pi i\pm\ln\left((y^{2}-1)^{% 1/2}+y\right),$ $y\in[1,\infty)$,
 4.37.18 $\operatorname{arcsinh}\left(iy\right)=-\tfrac{1}{2}\pi i\pm\ln\left((y^{2}-1)^% {1/2}-y\right),$ $y\in(-\infty,-1]$,

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

### Inverse Hyperbolic Cosine

 4.37.19 $\operatorname{arccosh}z=\ln\left(\pm(z^{2}-1)^{1/2}+z\right),$ $z\in\mathbb{C}\setminus(-\infty,1)$,

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

 4.37.20 $\operatorname{arccosh}\left(\mathrm{i}y\right)=\pm\tfrac{1}{2}\pi\mathrm{i}+% \ln\left((y^{2}+1)^{1/2}\pm y\right),$ $y\gtrless 0$.

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 $\operatorname{arccosh}z=2\ln\left(\left(\frac{z+1}{2}\right)^{1/2}+\left(\frac% {z-1}{2}\right)^{1/2}\right),$ $z\in\mathbb{C}\setminus(-\infty,1)$;

see Kahan (1987).

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

 4.37.22 $\operatorname{arccosh}x=\pm\ln\left(i(1-x^{2})^{1/2}+x\right),$ $x\in(-1,1]$,

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

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

 4.37.23 $\operatorname{arccosh}x=\pm\pi i+\ln\left((x^{2}-1)^{1/2}-x\right),$ $x\in(-\infty,-1]$,

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

### Inverse Hyperbolic Tangent

 4.37.24 $\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right),$ $z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$;

compare Figure 4.37.1(iii). On the cuts

 4.37.25 $\operatorname{arctanh}x=\pm\tfrac{1}{2}\pi i+\tfrac{1}{2}\ln\left(\frac{x+1}{x% -1}\right),$ $x\in(-\infty,-1)\cup(1,\infty)$,

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

### Other Inverse Functions

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

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 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 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 4

are respectively given by

 4.37.29 $\displaystyle w$ $\displaystyle=\operatorname{Arcsinh}z=(-1)^{k}\operatorname{arcsinh}z+k\pi i,$ 4.37.30 $\displaystyle w$ $\displaystyle=\operatorname{Arccosh}z=\pm\operatorname{arccosh}z+2k\pi i,$ 4.37.31 $\displaystyle w$ $\displaystyle=\operatorname{Arctanh}z=\operatorname{arctanh}z+k\pi i,$ $z\neq\pm 1$.

## §4.37(vi) Interrelations

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