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

§4.37 Inverse Hyperbolic Functions

Keywords:: inverse hyperbolic functions
Referenced by:: §4.2(i)
Permalink:: http://dlmf.nist.gov/4.37

§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

Keywords:: inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.37.i

The general values of the inverse hyperbolic functions are defined by

4.37.1 $\mathop{\mathrm{Arcsinh}\/}\nolimits z=\int_{0}^{z}\frac{dt}{(1+t^{2})^{{1/2}}},$

Defines:: $\mathop{\mathrm{Arcsinh}\/}\nolimits z$ : general inverse hyperbolic sine function
Symbols:: : differential of , $\int$ : integral and : complex variable
Referenced by:: §4.37(i), §4.37(ii)
Permalink:: http://dlmf.nist.gov/4.37.E1
Encodings:: TeX, pMML, png

4.37.2 $\mathop{\mathrm{Arccosh}\/}\nolimits z=\int_{1}^{z}\frac{dt}{(t^{2}-1)^{{1/2}}},$

Defines:: $\mathop{\mathrm{Arccosh}\/}\nolimits z$ : general inverse hyperbolic cosine function
Symbols:: : differential of , $\int$ : integral and : complex variable
Referenced by:: §4.37(i)
Permalink:: http://dlmf.nist.gov/4.37.E2
Encodings:: TeX, pMML, png

4.37.3 $\mathop{\mathrm{Arctanh}\/}\nolimits z=\int_{0}^{z}\frac{dt}{1-t^{2}},$ $z\neq\pm 1$ ,

Defines:: $\mathop{\mathrm{Arctanh}\/}\nolimits z$ : general inverse hyperbolic tangent function
Symbols:: : differential of , $\int$ : integral and : complex variable
Referenced by:: §4.37(i), §4.37(ii)
Permalink:: http://dlmf.nist.gov/4.37.E3
Encodings:: TeX, pMML, png

4.37.4 $\mathop{\mathrm{Arccsch}\/}\nolimits z=\mathop{\mathrm{Arcsinh}\/}\nolimits\!% \left(1/z\right),$

Defines:: $\mathop{\mathrm{Arccsch}\/}\nolimits z$ : general inverse hyperbolic cosecant function
Symbols:: $\mathop{\mathrm{Arcsinh}\/}\nolimits z$ : general inverse hyperbolic sine function and : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E4
Encodings:: TeX, pMML, png

4.37.5 $\mathop{\mathrm{Arcsech}\/}\nolimits z=\mathop{\mathrm{Arccosh}\/}\nolimits\!% \left(1/z\right),$

Defines:: $\mathop{\mathrm{Arcsech}\/}\nolimits z$ : general inverse hyperbolic secant function
Symbols:: $\mathop{\mathrm{Arccosh}\/}\nolimits z$ : general inverse hyperbolic cosine function and : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E5
Encodings:: TeX, pMML, png

4.37.6 $\mathop{\mathrm{Arccoth}\/}\nolimits z=\mathop{\mathrm{Arctanh}\/}\nolimits\!% \left(1/z\right).$

Defines:: $\mathop{\mathrm{Arccoth}\/}\nolimits z$ : general inverse hyperbolic cotangent function
Symbols:: $\mathop{\mathrm{Arctanh}\/}\nolimits z$ : general inverse hyperbolic tangent function and : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E6
Encodings:: TeX, pMML, png

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 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 . $\mathop{\mathrm{Arcsinh}\/}\nolimits z$ and $\mathop{\mathrm{Arccsch}\/}\nolimits z$ have branch points at $z=\pm i$ ; the other four functions have branch points at $z=\pm 1$ .

§4.37(ii) Principal Values

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

The principal values (or principal branches) of the inverse $\mathop{\sinh\/}\nolimits$ , $\mathop{\cosh\/}\nolimits$ , and $\mathop{\tanh\/}\nolimits$ are obtained by introducing cuts in the -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 $\mathop{\mathrm{arcsinh}\/}\nolimits$ , $\mathop{\mathrm{arccosh}\/}\nolimits$ , $\mathop{\mathrm{arctanh}\/}\nolimits$ 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 $\mathop{\mathrm{arccsch}\/}\nolimits z=\mathop{\mathrm{arcsinh}\/}\nolimits\!% \left(1/z\right),$

Defines:: $\mathop{\mathrm{arccsch}\/}\nolimits z$ : inverse hyperbolic cosecant function
Symbols:: $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : inverse hyperbolic sine function and : 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

4.37.8 $\mathop{\mathrm{arcsech}\/}\nolimits z=\mathop{\mathrm{arccosh}\/}\nolimits\!% \left(1/z\right).$

Defines:: $\mathop{\mathrm{arcsech}\/}\nolimits z$ : inverse hyperbolic secant function
Symbols:: $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function and : complex variable
A&S Ref:: 4.6.5
Permalink:: http://dlmf.nist.gov/4.37.E8
Encodings:: TeX, pMML, png

4.37.9 $\mathop{\mathrm{arccoth}\/}\nolimits z=\mathop{\mathrm{arctanh}\/}\nolimits\!% \left(1/z\right),$ $z\neq\pm 1$ .

Defines:: $\mathop{\mathrm{arccoth}\/}\nolimits z$ : inverse hyperbolic cotangent function
Symbols:: $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function and : 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

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.


(i) $\mathop{\mathrm{arcsinh}\/}\nolimits z$	(ii) $\mathop{\mathrm{arccosh}\/}\nolimits z$	(iii) $\mathop{\mathrm{arctanh}\/}\nolimits z$

(iv) $\mathop{\mathrm{arccsch}\/}\nolimits z$	(v) $\mathop{\mathrm{arcsech}\/}\nolimits z$	(vi) $\mathop{\mathrm{arccoth}\/}\nolimits z$

Figure 4.37.1:

-plane. Branch cuts for the inverse hyperbolic functions.

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

Notes:: (4.37.11) follows from (4.37.19).
Keywords:: inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.37.iii

4.37.10 $\mathop{\mathrm{arcsinh}\/}\nolimits\!\left(-z\right)=-\mathop{\mathrm{arcsinh% }\/}\nolimits z.$

Symbols:: $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : inverse hyperbolic sine function and : complex variable
A&S Ref:: 4.6.11
Permalink:: http://dlmf.nist.gov/4.37.E10
Encodings:: TeX, pMML, png

4.37.11 $\mathop{\mathrm{arccosh}\/}\nolimits\!\left(-z\right)=\pm\pi i+\mathop{\mathrm% {arccosh}\/}\nolimits z,$ $\imagpart{z}\gtrless 0$ .

Symbols:: $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\imagpart{}$ : imaginary part and : 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

4.37.12 $\mathop{\mathrm{arctanh}\/}\nolimits\!\left(-z\right)=-\mathop{\mathrm{arctanh% }\/}\nolimits z,$ $z\neq\pm 1$ .

Symbols:: $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function and : complex variable
A&S Ref:: 4.6.13
Permalink:: http://dlmf.nist.gov/4.37.E12
Encodings:: TeX, pMML, png

4.37.13 $\mathop{\mathrm{arccsch}\/}\nolimits\!\left(-z\right)=-\mathop{\mathrm{arccsch% }\/}\nolimits z.$

Symbols:: $\mathop{\mathrm{arccsch}\/}\nolimits z$ : inverse hyperbolic cosecant function and : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E13
Encodings:: TeX, pMML, png

4.37.14 $\mathop{\mathrm{arcsech}\/}\nolimits\!\left(-z\right)=\mp\pi i+\mathop{\mathrm% {arcsech}\/}\nolimits z,$ $\imagpart{z}\gtrless 0$ .

Symbols:: $\mathop{\mathrm{arcsech}\/}\nolimits z$ : inverse hyperbolic secant function, $\imagpart{}$ : imaginary part and : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E14
Encodings:: TeX, pMML, png

4.37.15 $\mathop{\mathrm{arccoth}\/}\nolimits\!\left(-z\right)=-\mathop{\mathrm{arccoth% }\/}\nolimits z,$ $z\neq\pm 1$ .

Symbols:: $\mathop{\mathrm{arccoth}\/}\nolimits z$ : inverse hyperbolic cotangent function and : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E15
Encodings:: TeX, pMML, png

§4.37(iv) Logarithmic Forms

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).
Keywords:: inverse hyperbolic functions
Referenced by:: §4.38(ii), ¶ ‣ §4.45(i), §4.45(ii)
Permalink:: http://dlmf.nist.gov/4.37.iv

Throughout this subsection all quantities assume their principal values.

¶ Inverse Hyperbolic Sine

4.37.16 $\mathop{\mathrm{arcsinh}\/}\nolimits z=\mathop{\ln\/}\nolimits\!\left((z^{2}+1% )^{{1/2}}+z\right),$ $z/i\in\Complex\setminus(-\infty,-1)\cup(1,\infty)$ ;

Symbols:: $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : inverse hyperbolic sine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : open interval, $\setminus$ : set subtraction, $\cup$ : union and : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E16
Encodings:: TeX, pMML, png

compare Figure 4.37.1(i). On the cuts

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

Symbols:: : half-closed interval, $\in$ : element of, $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : inverse hyperbolic sine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : real variable
Permalink:: http://dlmf.nist.gov/4.37.E17
Encodings:: TeX, pMML, png

4.37.18 $\mathop{\mathrm{arcsinh}\/}\nolimits\!\left(iy\right)=-\tfrac{1}{2}\pi i\pm% \mathop{\ln\/}\nolimits\!\left((y^{2}-1)^{{1/2}}-y\right),$ $y\in(-\infty,-1]$ ,

Symbols:: $\in$ : element of, $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : inverse hyperbolic sine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : half-closed interval and : real variable
Permalink:: http://dlmf.nist.gov/4.37.E18
Encodings:: TeX, pMML, png

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

¶ Inverse Hyperbolic Cosine

4.37.19 $\mathop{\mathrm{arccosh}\/}\nolimits z=\mathop{\ln\/}\nolimits\!\left(\pm(z^{2% }-1)^{{1/2}}+z\right),$ $z\in\Complex\setminus(-\infty,1)$ ,

Symbols:: $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : open interval, $\setminus$ : set subtraction and : 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

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

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

Symbols:: $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : real variable
Referenced by:: ¶ ‣ §4.37(iv)
Permalink:: http://dlmf.nist.gov/4.37.E20
Encodings:: TeX, pMML, png

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 $\mathop{\mathrm{arccosh}\/}\nolimits z=2\mathop{\ln\/}\nolimits\!\left(\left(% \frac{z+1}{2}\right)^{{1/2}}+\left(\frac{z-1}{2}\right)^{{1/2}}\right),$ $z\in\Complex\setminus(-\infty,1)$ ;

Symbols:: $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : open interval, $\setminus$ : set subtraction and : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E21
Encodings:: TeX, pMML, png

see Kahan (1987).

On the part of the cuts from −1 to 1

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

Symbols:: $\in$ : element of, $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : half-closed interval and : real variable
Permalink:: http://dlmf.nist.gov/4.37.E22
Encodings:: TeX, pMML, png

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

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

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

Symbols:: $\in$ : element of, $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : half-closed interval and : real variable
Permalink:: http://dlmf.nist.gov/4.37.E23
Encodings:: TeX, pMML, png

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

¶ Inverse Hyperbolic Tangent

4.37.24 $\mathop{\mathrm{arctanh}\/}\nolimits z=\tfrac{1}{2}\mathop{\ln\/}\nolimits\!% \left(\frac{1+z}{1-z}\right),$ $z\in\Complex\setminus(-\infty,-1]\cup[1,\infty)$ ;

Symbols:: : half-closed interval, $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : half-closed interval, $\setminus$ : set subtraction, $\cup$ : union and : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E24
Encodings:: TeX, pMML, png

compare Figure 4.37.1(iii). On the cuts

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

Symbols:: $\in$ : element of, $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : open interval, $\cup$ : union and : real variable
Permalink:: http://dlmf.nist.gov/4.37.E25
Encodings:: TeX, pMML, png

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

¶ Other Inverse Functions

Keywords:: inverse hyperbolic functions

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

§4.37(v) Fundamental Property

Keywords:: inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.37.v

With $k\in\Integer$ , the general solutions of the equations

4.37.26 $z=\mathop{\sinh\/}\nolimits w,$

Symbols:: $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E26
Encodings:: TeX, pMML, png

4.37.27 $z=\mathop{\cosh\/}\nolimits w,$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E27
Encodings:: TeX, pMML, png

4.37.28 $z=\mathop{\tanh\/}\nolimits w,$

Symbols:: $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E28
Encodings:: TeX, pMML, png

are respectively given by

4.37.29 $w=\mathop{\mathrm{Arcsinh}\/}\nolimits z=(-1)^{k}\mathop{\mathrm{arcsinh}\/}% \nolimits z+k\pi i,$

Symbols:: $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : inverse hyperbolic sine function, $\mathop{\mathrm{Arcsinh}\/}\nolimits z$ : general inverse hyperbolic sine function, : integer and : complex variable
A&S Ref:: 4.6.8
Permalink:: http://dlmf.nist.gov/4.37.E29
Encodings:: TeX, pMML, png

4.37.30 $w=\mathop{\mathrm{Arccosh}\/}\nolimits z=\pm\mathop{\mathrm{arccosh}\/}% \nolimits z+2k\pi i,$

Symbols:: $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\mathop{\mathrm{Arccosh}\/}\nolimits z$ : general inverse hyperbolic cosine function, : integer and : complex variable
A&S Ref:: 4.6.9
Permalink:: http://dlmf.nist.gov/4.37.E30
Encodings:: TeX, pMML, png

4.37.31 $w=\mathop{\mathrm{Arctanh}\/}\nolimits z=\mathop{\mathrm{arctanh}\/}\nolimits z% +k\pi i,$ $z\neq\pm 1$ .

Symbols:: $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function, $\mathop{\mathrm{Arctanh}\/}\nolimits z$ : general inverse hyperbolic tangent function, : integer and : complex variable
A&S Ref:: 4.6.10
Permalink:: http://dlmf.nist.gov/4.37.E31
Encodings:: TeX, pMML, png

§4.37(vi) Interrelations

Keywords:: inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.37.vi

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 4.36 Infinite Products and Partial Fractions 4.38 Inverse Hyperbolic Functions: Further Properties