Inverse hyperbolic functions

Definitions in terms of logarithms

Since the hyperbolic functions are quadratic rational functions of the exponential function ${\displaystyle \exp x,}$ they may be solved using the quadratic formula and then written in terms of the natural logarithm.

$${\displaystyle \begin{array}{cccc}\mathrm{arsinh}x& =\ln \left(x+\sqrt{x^2+1}\right)& -\mathrm{\infty }& <x<\mathrm{\infty },\\ \mathrm{arcosh}x& =\ln \left(x+\sqrt{x^2-1}\right)& 1& \le x<\mathrm{\infty },\\ \mathrm{artanh}x& =\frac{1}{2}\ln \frac{1+x}{1-x}& -1& <x<1,\\ \mathrm{arcsch}x& =\ln \left(\frac{1}{x}+\sqrt{\frac{1}{x^2}+1}\right)& -\mathrm{\infty }& <x<\mathrm{\infty },x\ne 0,\\ \mathrm{arsech}x& =\ln \left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)& 0& <x\le 1,\\ \mathrm{arcoth}x& =\frac{1}{2}\ln \frac{x+1}{x-1}& -\mathrm{\infty }& <x<-1\text{or}1<x<\mathrm{\infty }.\end{array}}$$

For complex arguments, the inverse circular and hyperbolic functions, the square root, and the natural logarithm are all multi-valued functions.

Addition formulae

$${\displaystyle \mathrm{arsinh}u\pm \mathrm{arsinh}v=\mathrm{arsinh}\left(u\sqrt{1+v^2}\pm v\sqrt{1+u^2}\right)}$$ $${\displaystyle \mathrm{arcosh}u\pm \mathrm{arcosh}v=\mathrm{arcosh}\left(uv\pm \sqrt{(u^2-1)(v^2-1)}\right)}$$ $${\displaystyle \mathrm{artanh}u\pm \mathrm{artanh}v=\mathrm{artanh}\left(\frac{u\pm v}{1\pm uv}\right)}$$ $${\displaystyle \mathrm{arcoth}u\pm \mathrm{arcoth}v=\mathrm{arcoth}\left(\frac{1\pm uv}{u\pm v}\right)}$$ $${\displaystyle \begin{array}{cc}\mathrm{arsinh}u+\mathrm{arcosh}v& =\mathrm{arsinh}\left(uv+\sqrt{(1+u^2)(v^2-1)}\right)\\ & =\mathrm{arcosh}\left(v\sqrt{1+u^2}+u\sqrt{v^2-1}\right)\end{array}}$$

Other identities

$${\displaystyle \begin{array}{ccc}2\mathrm{arcosh}x& =\mathrm{arcosh}(2x^2-1)& \text{ for }x\ge 1\\ 4\mathrm{arcosh}x& =\mathrm{arcosh}(8x^4-8x^2+1)& \text{ for }x\ge 1\\ 2\mathrm{arsinh}x& =\pm \mathrm{arcosh}(2x^2+1)\\ 4\mathrm{arsinh}x& =\mathrm{arcosh}(8x^4+8x^2+1)& \text{ for }x\ge 0\end{array}}$$

$${\displaystyle \ln (x)=\mathrm{arcosh}\left(\frac{x^2+1}{2x}\right)=\mathrm{arsinh}\left(\frac{x^2-1}{2x}\right)=\mathrm{artanh}\left(\frac{x^2-1}{x^2+1}\right)}$$

Composition of hyperbolic and inverse hyperbolic functions

$${\displaystyle \begin{array}{cc}& \sinh (\mathrm{arcosh}x)=\sqrt{x^2-1}\text{for}|x|>1\\ & \sinh (\mathrm{artanh}x)=\frac{x}{\sqrt{1-x^2}}\text{for}-1<x<1\\ & \cosh (\mathrm{arsinh}x)=\sqrt{1+x^2}\\ & \cosh (\mathrm{artanh}x)=\frac{1}{\sqrt{1-x^2}}\text{for}-1<x<1\\ & \tanh (\mathrm{arsinh}x)=\frac{x}{\sqrt{1+x^2}}\\ & \tanh (\mathrm{arcosh}x)=\frac{\sqrt{x^2-1}}{x}\text{for}|x|>1\end{array}}$$

Composition of inverse hyperbolic and circular functions

$${\displaystyle \mathrm{arsinh}\left(\tan \alpha \right)=\mathrm{artanh}\left(\sin \alpha \right)=\ln \left(\frac{1+\sin \alpha }{\cos \alpha }\right)=\pm \mathrm{arcosh}\left(\frac{1}{\cos \alpha }\right)}$$

${\displaystyle \ln \left(\left|\tan \alpha \right|\right)=-\mathrm{artanh}\left(\cos 2\alpha \right)}$

Conversions

$${\displaystyle \ln x=\mathrm{artanh}\left(\frac{x^2-1}{x^2+1}\right)=\mathrm{arsinh}\left(\frac{x^2-1}{2x}\right)=\mathrm{sgn}(x-1)\mathrm{arcosh}\left(\frac{x^2+1}{2x}\right)}$$

$${\displaystyle \mathrm{artanh}x=\mathrm{arsinh}\left(\frac{x}{\sqrt{1-x^2}}\right)=\mathrm{sgn}x\mathrm{arcosh}\left(\frac{1}{\sqrt{1-x^2}}\right)}$$

$${\displaystyle \mathrm{arsinh}x=\mathrm{artanh}\left(\frac{x}{\sqrt{1+x^2}}\right)=\mathrm{sgn}x\mathrm{arcosh}\left(\sqrt{1+x^2}\right)}$$

$${\displaystyle \mathrm{arcosh}x=\left|\mathrm{arsinh}\left(\sqrt{x^2-1}\right)\right|=\left|\mathrm{artanh}\left(\frac{\sqrt{x^2-1}}{x}\right)\right|}$$

Derivatives

$${\displaystyle \begin{array}{cc}\frac{d}{dx}\mathrm{arsinh}x& =\frac{1}{\sqrt{x^2+1}},\text{ for all real }x\\ \frac{d}{dx}\mathrm{arcosh}x& =\frac{1}{\sqrt{x^2-1}},\text{ for all real }x>1\\ \frac{d}{dx}\mathrm{artanh}x& =\frac{1}{1-x^2},\text{ for all real }|x|<1\\ \frac{d}{dx}\mathrm{arcoth}x& =\frac{1}{1-x^2},\text{ for all real }|x|>1\\ \frac{d}{dx}\mathrm{arsech}x& =\frac{-1}{x\sqrt{1-x^2}},\text{ for all real }x\in (0,1)\\ \frac{d}{dx}\mathrm{arcsch}x& =\frac{-1}{|x|\sqrt{1+x^2}},\text{ for all real }x\text{, except }0\end{array}}$$

These formulas can be derived in terms of the derivatives of hyperbolic functions. For example, if ${\displaystyle x=\sinh \theta }$, then $dx/d\theta =\cosh \theta =\sqrt{1+x^2},$ so $${\displaystyle \frac{d}{dx}\mathrm{arsinh}(x)=\frac{d\theta }{dx}=\frac{1}{dx/d\theta }=\frac{1}{\sqrt{1+x^2}}.}$$

Series expansions

Expansion series can be obtained for the above functions:

$${\displaystyle \begin{array}{cc}\mathrm{arsinh}x& =x-\left(\frac{1}{2}\right)\frac{x^3}{3}+\left(\frac{1\cdot 3}{2\cdot 4}\right)\frac{x^5}{5}-\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)\frac{x^7}{7}\pm \cdots \\ & =\sum _{n=0}^{\mathrm{\infty }}\left(\frac{(-1{)}^n(2n)!}{2^{2n}(n!{)}^2}\right)\frac{x^{2n+1}}{2n+1},\left|x\right|<1\end{array}}$$

$${\displaystyle \begin{array}{cc}\mathrm{arcosh}x& =\ln (2x)-\left(\left(\frac{1}{2}\right)\frac{x^{-2}}{2}+\left(\frac{1\cdot 3}{2\cdot 4}\right)\frac{x^{-4}}{4}+\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)\frac{x^{-6}}{6}+\cdots \right)\\ & =\ln (2x)-\sum _{n=1}^{\mathrm{\infty }}\left(\frac{(2n)!}{2^{2n}(n!{)}^2}\right)\frac{x^{-2n}}{2n},\left|x\right|>1\end{array}}$$

$${\displaystyle \begin{array}{cc}\mathrm{artanh}x& =x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+\cdots \\ & =\sum _{n=0}^{\mathrm{\infty }}\frac{x^{2n+1}}{2n+1},\left|x\right|<1\end{array}}$$

$${\displaystyle \begin{array}{cc}\mathrm{arcsch}x=\mathrm{arsinh}\frac{1}{x}& =x^{-1}-\left(\frac{1}{2}\right)\frac{x^{-3}}{3}+\left(\frac{1\cdot 3}{2\cdot 4}\right)\frac{x^{-5}}{5}-\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)\frac{x^{-7}}{7}\pm \cdots \\ & =\sum _{n=0}^{\mathrm{\infty }}\left(\frac{(-1{)}^n(2n)!}{2^{2n}(n!{)}^2}\right)\frac{x^{-(2n+1)}}{2n+1},\left|x\right|>1\end{array}}$$

$${\displaystyle \begin{array}{cc}\mathrm{arsech}x=\mathrm{arcosh}\frac{1}{x}& =\ln \frac{2}{x}-\left(\left(\frac{1}{2}\right)\frac{x^2}{2}+\left(\frac{1\cdot 3}{2\cdot 4}\right)\frac{x^4}{4}+\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)\frac{x^6}{6}+\cdots \right)\\ & =\ln \frac{2}{x}-\sum _{n=1}^{\mathrm{\infty }}\left(\frac{(2n)!}{2^{2n}(n!{)}^2}\right)\frac{x^{2n}}{2n},0<x\le 1\end{array}}$$

$${\displaystyle \begin{array}{cc}\mathrm{arcoth}x=\mathrm{artanh}\frac{1}{x}& =x^{-1}+\frac{x^{-3}}{3}+\frac{x^{-5}}{5}+\frac{x^{-7}}{7}+\cdots \\ & =\sum _{n=0}^{\mathrm{\infty }}\frac{x^{-(2n+1)}}{2n+1},\left|x\right|>1\end{array}}$$ An asymptotic expansion for arsinh is given by

$${\displaystyle \mathrm{arsinh}x=\ln (2x)+\sum _{n=1}^{\mathrm{\infty }}{\left(-1\right)}^{n-1}\frac{\left(2n-1\right)!!}{2n\left(2n\right)!!}\frac{1}{x^{2n}}}$$

Principal values in the complex plane

As functions of a complex variable, inverse hyperbolic functions are multivalued functions that are analytic except at a finite number of points. For such a function, it is common to define a principal value, which is a single valued analytic function which coincides with one specific branch of the multivalued function, over a domain consisting of the complex plane in which a finite number of arcs (usually half lines or line segments) have been removed. These arcs are called branch cuts. The principal value of the multifunction is chosen at a particular point and values elsewhere in the domain of definition are defined to agree with those found by analytic continuation.

For example, for the square root, the principal value is defined as the square root that has a positive real part. This defines a single valued analytic function, which is defined everywhere, except for non-positive real values of the variables (where the two square roots have a zero real part). This principal value of the square root function is denoted ${\displaystyle \sqrt{x}}$ in what follows. Similarly, the principal value of the logarithm, denoted ${\displaystyle \mathrm{Log}}$ in what follows, is defined as the value for which the imaginary part has the smallest absolute value. It is defined everywhere except for non-positive real values of the variable, for which two different values of the logarithm reach the minimum.

For all inverse hyperbolic functions, the principal value may be defined in terms of principal values of the square root and the logarithm function. However, in some cases, the formulas of § Definitions in terms of logarithms do not give a correct principal value, as giving a domain of definition which is too small and, in one case non-connected.

Principal value of the inverse hyperbolic sine

The principal value of the inverse hyperbolic sine is given by $${\displaystyle \mathrm{arsinh}z=\mathrm{Log}(z+\sqrt{z^2+1}).}$$

The argument of the square root is a non-positive real number, if and only if z belongs to one of the intervals [i, +i∞) and (−i∞, −i] of the imaginary axis. If the argument of the logarithm is real, then it is positive. Thus this formula defines a principal value for arsinh, with branch cuts [i, +i∞) and (−i∞, −i]. This is optimal, as the branch cuts must connect the singular points i and −i to infinity.

Principal value of the inverse hyperbolic cosine

The formula for the inverse hyperbolic cosine given in § Inverse hyperbolic cosine is not convenient, since similar to the principal values of the logarithm and the square root, the principal value of arcosh would not be defined for imaginary z. Thus the square root has to be factorized, leading to $${\displaystyle \mathrm{arcosh}z=\mathrm{Log}(z+\sqrt{z+1}\sqrt{z-1}).}$$

The principal values of the square roots are both defined, except if z belongs to the real interval (−∞, 1]. If the argument of the logarithm is real, then z is real and has the same sign. Thus, the above formula defines a principal value of arcosh outside the real interval (−∞, 1], which is thus the unique branch cut.

Principal values of the inverse hyperbolic tangent and cotangent

The formulas given in § Definitions in terms of logarithms suggests $${\displaystyle \begin{array}{cc}\mathrm{artanh}z& =\frac{1}{2}\mathrm{Log}\left(\frac{1+z}{1-z}\right)\\ \mathrm{arcoth}z& =\frac{1}{2}\mathrm{Log}\left(\frac{z+1}{z-1}\right)\end{array}}$$ for the definition of the principal values of the inverse hyperbolic tangent and cotangent. In these formulas, the argument of the logarithm is real if and only if z is real. For artanh, this argument is in the real interval (−∞, 0], if z belongs either to (−∞, −1] or to [1, ∞). For arcoth, the argument of the logarithm is in (−∞, 0], if and only if z belongs to the real interval [−1, 1].

Therefore, these formulas define convenient principal values, for which the branch cuts are (−∞, −1] and [1, ∞) for the inverse hyperbolic tangent, and [−1, 1] for the inverse hyperbolic cotangent.

In view of a better numerical evaluation near the branch cuts, some authors use the following definitions of the principal values, although the second one introduces a removable singularity at z = 0. The two definitions of ${\displaystyle \mathrm{artanh}}$ differ for real values of z with z > 1. The ones of ${\displaystyle \mathrm{arcoth}}$ differ for real values of z with z ∈ [0, 1). $${\displaystyle \begin{array}{cc}\mathrm{artanh}z& =\frac{1}{2}\mathrm{Log}\left(1+z\right)-\frac{1}{2}\mathrm{Log}\left(1-z\right)\\ \mathrm{arcoth}z& =\frac{1}{2}\mathrm{Log}\left(1+\frac{1}{z}\right)-\frac{1}{2}\mathrm{Log}\left(1-\frac{1}{z}\right)\end{array}}$$

Principal value of the inverse hyperbolic cosecant

For the inverse hyperbolic cosecant, the principal value is defined as $${\displaystyle \mathrm{arcsch}z=\mathrm{Log}\left(\frac{1}{z}+\sqrt{\frac{1}{z^2}+1}\right).}$$

It is defined except when the arguments of the logarithm and the square root are non-positive real numbers. The principal value of the square root is thus defined outside the interval [−i, i] of the imaginary line. If the argument of the logarithm is real, then z is a non-zero real number, and this implies that the argument of the logarithm is positive.

Thus, the principal value is defined by the above formula outside the branch cut, consisting of the interval [−i, i] of the imaginary line.

(At z = 0, there is a singular point that is included in the branch cut.)

Principal value of the inverse hyperbolic secant

Here, as in the case of the inverse hyperbolic cosine, we have to factorize the square root. This gives the principal value $${\displaystyle \mathrm{arsech}z=\mathrm{Log}\left(\frac{1}{z}+\sqrt{\frac{1}{z}+1}\sqrt{\frac{1}{z}-1}\right).}$$

If the argument of a square root is real, then z is real, and it follows that both principal values of square roots are defined, except if z is real and belongs to one of the intervals (−∞, 0] and [1, +∞). If the argument of the logarithm is real and negative, then z is also real and negative. It follows that the principal value of arsech is well defined, by the above formula outside two branch cuts, the real intervals (−∞, 0] and [1, +∞).

For z = 0, there is a singular point that is included in one of the branch cuts.

Graphical representation

In the following graphical representation of the principal values of the inverse hyperbolic functions, the branch cuts appear as discontinuities of the color. The fact that the whole branch cuts appear as discontinuities, shows that these principal values may not be extended into analytic functions defined over larger domains. In other words, the above defined branch cuts are minimal.