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Inverse hyperbolic functions

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Mathematical functions

Graphs of the inverse hyperbolic functions

The hyperbolic functionssinh,cosh, andtanh with respect to a unit hyperbola are analogous to circular functionssin,cos,tan with respect to a unit circle. The argument to the hyperbolic functions is a hyperbolic angle measure.

Inmathematics, theinverse hyperbolic functions areinverses of thehyperbolic functions, analogous to theinverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions, prefixed witharc- orar- or with a superscript ${-1}$ (for examplearcsinh,arsinh, or $\sinh ^{-1}$ ).

For a given value of a hyperbolic function, the inverse hyperbolic function provides the correspondinghyperbolic angle measure, for example $\operatorname {arsinh} (\sinh a)=a$ and $\sinh(\operatorname {arsinh} x)=x.$ Hyperbolic angle measure is thelength of an arc of aunit hyperbola $x^{2}-y^{2}=1$ as measured in the Lorentzian plane (not the length of a hyperbolic arc in theEuclidean plane), and twice thearea of the correspondinghyperbolic sector. This is analogous to the waycircular angle measure is the arc length of an arc of theunit circle in the Euclidean plane or twice the area of the correspondingcircular sector. Alternately hyperbolic angle is the area of a sector of the hyperbola $xy=1.$ Some authors call the inverse hyperbolic functionshyperbolic area functions.^[1]

Hyperbolic functions occur in the calculation of angles and distances inhyperbolic geometry. They also occur in the solutions of many lineardifferential equations (such as the equation defining acatenary),cubic equations, andLaplace's equation inCartesian coordinates.Laplace's equations are important in many areas ofphysics, includingelectromagnetic theory,heat transfer,fluid dynamics, andspecial relativity.

Notation

A ray through theunit hyperbolax² −y² = 1 in the point(cosha, sinha), wherea is twice the area between the ray, the hyperbola, and thex-axis

The earliest and most widely adopted symbols use the prefixarc- (that is:arcsinh,arccosh,arctanh,arcsech,arccsch,arccoth), by analogy with theinverse circular functions (arcsin, etc.). For aunit hyperbola ("Lorentzian circle") in the Lorentzian plane (pseudo-Euclidean plane ofsignature(1, 1))^[2] or in thehyperbolic number plane,^[3] thehyperbolic angle measure (argument to the hyperbolic functions) is indeed thearc length of a hyperbolic arc.

Also common is the notation $\sinh ^{-1},$ $\cosh ^{-1},$ etc.,^[4]^[5] although care must be taken to avoid misinterpretations of the superscript −1 as an exponent. The standard convention is that $\sinh ^{-1}x$ or $\sinh ^{-1}(x)$ means the inverse function while $(\sinh x)^{-1}$ or $\sinh(x)^{-1}$ means thereciprocal $1/\sinh x.$ Especially inconsistent is the conventional use of positiveinteger superscripts to indicate an exponent rather than function composition, e.g. $\sinh ^{2}x$ conventionally means $(\sinh x)^{2}$ andnot $\sinh(\sinh x).$

Because the argument of hyperbolic functions isnot the arc length of a hyperbolic arc in theEuclidean plane, some authors have condemned the prefixarc-, arguing that the prefixar- (for'area') orarg- (for'argument') should be preferred.^[6] Following this recommendation, theISO 80000-2 standard abbreviations use the prefixar- (that is:arsinh,arcosh,artanh,arsech,arcsch,arcoth).

In computer programming languages, inverse circular and hyperbolic functions are often named with the shorter prefixa- (asinh, etc.).

This article will consistently adopt the prefixar- for convenience.

Definitions in terms of logarithms

Since thehyperbolic functions are quadraticrational functions of the exponential function $\exp x,$ they may be solved using thequadratic formula and then written in terms of thenatural logarithm.

${\begin{aligned}\operatorname {arsinh} x&=\ln \left(x+{\sqrt {x^{2}+1}}\right)&-\infty &<x<\infty ,\\[10mu]\operatorname {arcosh} x&=\ln \left(x+{\sqrt {x^{2}-1}}\right)&1&\leq x<\infty ,\\[10mu]\operatorname {artanh} x&={\frac {1}{2}}\ln {\frac {1+x}{1-x}}&-1&<x<1,\\[10mu]\operatorname {arcsch} x&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}+1}}\right)&-\infty &<x<\infty ,\ x\neq 0,\\[10mu]\operatorname {arsech} x&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}-1}}\right)&0&<x\leq 1,\\[10mu]\operatorname {arcoth} x&={\frac {1}{2}}\ln {\frac {x+1}{x-1}}&-\infty &<x<-1\ \ {\text{or}}\ \ 1<x<\infty .\end{aligned}}$

Forcomplex arguments, the inverse circular and hyperbolic functions, thesquare root, and the natural logarithm are allmulti-valued functions.

Addition formulae

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

Other identities

${\begin{aligned}2\operatorname {arcosh} x&=\operatorname {arcosh} (2x^{2}-1)&\quad {\hbox{ for }}x\geq 1\\4\operatorname {arcosh} x&=\operatorname {arcosh} (8x^{4}-8x^{2}+1)&\quad {\hbox{ for }}x\geq 1\\2\operatorname {arsinh} x&=\pm \operatorname {arcosh} (2x^{2}+1)\\4\operatorname {arsinh} x&=\operatorname {arcosh} (8x^{4}+8x^{2}+1)&\quad {\hbox{ for }}x\geq 0\end{aligned}}$

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

Composition of hyperbolic and inverse hyperbolic functions

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

Composition of inverse hyperbolic and circular functions

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

$\ln \left(\left|\tan \alpha \right|\right)=-\operatorname {artanh} \left(\cos 2\alpha \right)$ ^[7]

Conversions

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

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

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

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

Derivatives

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

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

Series expansions

Expansion series can be obtained for the above functions:

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

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

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

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

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

${\begin{aligned}\operatorname {arcoth} x=\operatorname {artanh} {\frac {1}{x}}&=x^{-1}+{\frac {x^{-3}}{3}}+{\frac {x^{-5}}{5}}+{\frac {x^{-7}}{7}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {x^{-(2n+1)}}{2n+1}},\qquad \left|x\right|>1\end{aligned}}$ Anasymptotic expansion for arsinh is given by

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

Principal values in the complex plane

Asfunctions of a complex variable, inverse hyperbolic functions aremultivalued functions that areanalytic except at a finite number of points. For such a function, it is common to define aprincipal value, which is a single valued analytic function which coincides with one specific branch of the multivalued function, over a domain consisting of thecomplex plane in which a finite number ofarcs (usuallyhalf lines orline segments) have been removed. These arcs are calledbranch 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 byanalytic continuation.

For example, for the square root, the principal value is defined as the square root that has a positivereal 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 ${\sqrt {x}}$ in what follows. Similarly, the principal value of the logarithm, denoted $\operatorname {Log}$ in what follows, is defined as the value for which theimaginary 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 casenon-connected.

Principal value of the inverse hyperbolic sine

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

The argument of the square root is a non-positivereal number,if and only ifz 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 pointsi 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 imaginaryz. Thus the square root has to be factorized, leading to $\operatorname {arcosh} z=\operatorname {Log} (z+{\sqrt {z+1}}{\sqrt {z-1}}\,)\,.$

The principal values of the square roots are both defined, except ifz belongs to the real interval(−∞, 1]. If the argument of the logarithm is real, thenz 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 ${\begin{aligned}\operatorname {artanh} z&={\frac {1}{2}}\operatorname {Log} \left({\frac {1+z}{1-z}}\right)\\\operatorname {arcoth} z&={\frac {1}{2}}\operatorname {Log} \left({\frac {z+1}{z-1}}\right)\end{aligned}}$ 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 ifz is real. For artanh, this argument is in the real interval(−∞, 0], ifz belongs either to(−∞, −1] or to[1, ∞). For arcoth, the argument of the logarithm is in(−∞, 0], if and only ifz 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^{[citation needed]} use the following definitions of the principal values, although the second one introduces aremovable singularity atz = 0. The two definitions of $\operatorname {artanh}$ differ for real values ofz withz > 1. The ones of $\operatorname {arcoth}$ differ for real values ofz withz ∈ [0, 1). ${\begin{aligned}\operatorname {artanh} z&={\tfrac {1}{2}}\operatorname {Log} \left({1+z}\right)-{\tfrac {1}{2}}\operatorname {Log} \left({1-z}\right)\\\operatorname {arcoth} z&={\tfrac {1}{2}}\operatorname {Log} \left({1+{\frac {1}{z}}}\right)-{\tfrac {1}{2}}\operatorname {Log} \left({1-{\frac {1}{z}}}\right)\end{aligned}}$

Principal value of the inverse hyperbolic cosecant

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

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, thenz 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 thebranch cut, consisting of the interval[−i,i] of the imaginary line.

(Atz = 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 $\operatorname {arsech} z=\operatorname {Log} \left({\frac {1}{z}}+{\sqrt {{\frac {1}{z}}+1}}\,{\sqrt {{\frac {1}{z}}-1}}\right).$

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

Forz = 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 definedbranch cuts are minimal.

Square representing central portion of the complex z-plane painted in psychedelic colours

arsinh(z)

Square representing central portion of the complex z-plane painted in psychedelic colours

arcosh(z)

Square representing central portion of the complex z-plane painted in psychedelic colours

artanh(z)

Square representing central portion of the complex z-plane painted in psychedelic colours

arcoth(z)

Square representing central portion of the complex z-plane painted in psychedelic colours

arsech(z)

Square representing central portion of the complex z-plane painted in psychedelic colours

arcsch(z)

Inverse hyperbolic functions in the complexz-plane: the colour at each point in the planerepresents the complex value of the respective function at that point

See also

References

^For example:
Weltner, Klaus; et al. (2014) [2009].Mathematics for Physicists and Engineers (2nd ed.). Springer.ISBN 978-364254124-7.
Durán, Mario (2012).Mathematical methods for wave propagation in science and engineering. Vol. 1. Ediciones UC. p. 89.ISBN 9789561413146.
^Birman, Graciela S.; Nomizu, Katsumi (1984). "Trigonometry in Lorentzian Geometry".American Mathematical Monthly.91 (9):543–549.doi:10.1080/00029890.1984.11971490.JSTOR 2323737.
^Sobczyk, Garret (1995). "The hyperbolic number plane".College Mathematics Journal.26 (4):268–280.doi:10.1080/07468342.1995.11973712.
^Weisstein, Eric W."Inverse Hyperbolic Functions".Wolfram Mathworld. Retrieved2020-08-30.
"Inverse hyperbolic functions".Encyclopedia of Mathematics. Retrieved2020-08-30.
^Press, W.H.; Teukolsky, S.A.; Vetterling, WT; Flannery, B.P. (1992). "§ 5.6. Quadratic and Cubic Equations".Numerical Recipes in FORTRAN (2nd ed.). Cambridge University Press.ISBN 0-521-43064-X.
Woodhouse, N.M.J. (2003).Special Relativity. Springer. p. 71.ISBN 1-85233-426-6.
^Gullberg, Jan (1997).Mathematics: From the Birth of Numbers. W. W. Norton. p. 539.ISBN 039304002X.Another form of notation,arcsinhx,arccoshx, etc., is a practice to be condemned as these functions have nothing whatever to do witharc, but witharea, as is demonstrated by their full Latin names,arsinharea sinus hyperbolicus,arcosharea cosinus hyperbolicus, etc.
Zeidler, Eberhard;Hackbusch, Wolfgang; Schwarz, Hans Rudolf (2004). "§ 0.2.13 The inverse hyperbolic functions".Oxford Users' Guide to Mathematics. Translated by Hunt, Bruce. Oxford University Press. p. 68.ISBN 0198507631.The Latin names for the inverse hyperbolic functions arearea sinus hyperbolicus,area cosinus hyperbolicus,area tangens hyperbolicus andarea cotangens hyperbolicus (ofx).....
Zeidler & al. use the notationsarsinh, etc.; note that the quoted Latin names areback-formations, invented long afterNeo-Latin ceased to be in common use in mathematical literature.
Bronshtein, Ilja N.;Semendyayev, Konstantin A.; Musiol, Gerhard; Heiner, Mühlig (2007). "§ 2.10: Area Functions".Handbook of Mathematics (5th ed.). Springer. p. 91.doi:10.1007/978-3-540-72122-2.ISBN 978-3540721215.Thearea functions are the inverse functions of the hyperbolic functions, i.e., theinverse hyperbolic functions. The functionssinhx,tanhx, andcothx are strictly monotone, so they have unique inverses without any restriction; the functioncoshx has two monotonic intervals so we can consider two inverse functions. The namearea refers to the fact that the geometric definition of the functions is the area of certain hyperbolic sectors ...
Bacon, Harold Maile (1942).Differential and Integral Calculus. McGraw-Hill. p. 203.
^"Identities with inverse hyperbolic and trigonometric functions".math stackexchange.stackexchange. Retrieved3 November 2016.^{[user-generated source]}

Bibliography

Herbert Busemann and Paul J. Kelly (1953)Projective Geometry and Projective Metrics, page 207,Academic Press.

External links

"Inverse hyperbolic functions",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

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Inverse hyperbolic functions

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