Inverse hyperbolic functions
Inverse hyperbolic functions are the multiplicative inverses of the hyperbolic functions, consisting of six principal branches: the inverse hyperbolic sine (denoted \sinh^{-1} or \mathrm{arcsinh}), inverse hyperbolic cosine (\cosh^{-1} or \mathrm{arccosh}), inverse hyperbolic tangent (\tanh^{-1} or \mathrm{arctanh}), inverse hyperbolic cotangent (\coth^{-1} or \mathrm{arccoth}), inverse hyperbolic secant (\sech^{-1} or \mathrm{arcsech}), and inverse hyperbolic cosecant (\csch^{-1} or \mathrm{arccsch}).[1] These functions map real numbers within specified domains to real values, serving as essential tools in calculus for evaluating integrals of forms like \int \frac{1}{\sqrt{x^2 + 1}} \, dx = \sinh^{-1} x + C and in solving differential equations that model physical phenomena such as the shape of hanging chains (catenaries).[2][3] For real arguments, the inverse hyperbolic functions admit closed-form expressions using the natural logarithm, reflecting their close relationship to exponential functions via the definitions of sinh and cosh.[4] Specifically, \sinh^{-1} x = \ln\left(x + \sqrt{x^2 + 1}\right) for all real x, with domain \mathbb{R} and range \mathbb{R}; \cosh^{-1} x = \ln\left(x + \sqrt{x^2 - 1}\right) for x \geq 1, with domain [1, \infty) and range [0, \infty); and \tanh^{-1} x = \frac{1}{2} \ln\left(\frac{1 + x}{1 - x}\right) for -1 < x < 1, with domain (-1, 1) and range \mathbb{R}.[1] The remaining inverses follow analogous logarithmic forms: \coth^{-1} x = \frac{1}{2} \ln\left(\frac{x + 1}{x - 1}\right) for |x| > 1; \sech^{-1} x = \ln\left(\frac{1 + \sqrt{1 - x^2}}{x}\right) for $0 < x \leq 1; and \csch^{-1} x = \ln\left(\frac{1 + \sqrt{1 + x^2}}{|x|}\right) for x \neq 0.[1] These expressions ensure the functions are well-defined and single-valued over their principal branches, though they extend to multi-valued forms in the complex plane with branch cuts.[4] Key properties of inverse hyperbolic functions include their derivatives, which mirror the reciprocals of the hyperbolic functions evaluated at the inverse: for example, \frac{d}{dx} \sinh^{-1} x = \frac{1}{\sqrt{x^2 + 1}}, \frac{d}{dx} \cosh^{-1} x = \frac{1}{\sqrt{x^2 - 1}} for x > 1, and \frac{d}{dx} \tanh^{-1} x = \frac{1}{1 - x^2} for |x| < 1.[2] Similar derivative formulas hold for the other inverses, such as \frac{d}{dx} \sech^{-1} x = -\frac{1}{x \sqrt{1 - x^2}} for $0 < x < 1.[1] They also satisfy identities analogous to those of inverse trigonometric functions, including addition formulas and relations like \sinh^{-1} x = -i \sin^{-1}(i x) in the complex domain, facilitating computations and connections between hyperbolic and trigonometric analyses.[4] In applications, these functions model scenarios involving exponential growth or decay, such as the trajectory of particles in special relativity or the equilibrium shapes in mechanics, underscoring their utility beyond pure mathematics.[3]Introduction
Overview and motivation
The inverse hyperbolic functions are the mathematical inverses of the six hyperbolic functions, serving a role analogous to that of the inverse trigonometric functions for the trigonometric functions. Just as the inverse trigonometric functions invert the relationships defined by sine, cosine, and their reciprocals on the unit circle, the inverse hyperbolic functions invert the hyperbolic sine (sinh), hyperbolic cosine (cosh), hyperbolic tangent (tanh), and their reciprocals, which are intrinsically linked to the geometry of the hyperbola. The six inverse hyperbolic functions are denoted as arcsinh(x) or sinh⁻¹(x), arccosh(x) or cosh⁻¹(x), arctanh(x) or tanh⁻¹(x), arccoth(x) or coth⁻¹(x), arcsech(x) or sech⁻¹(x), and arccsch(x) or csch⁻¹(x).[5][6] These functions are multi-valued in the complex domain, requiring the selection of principal branches to define single-valued versions, but on the real line, they are typically defined to be real-valued over restricted domains where the corresponding hyperbolic functions are bijective. The domains and ranges for real arguments are as follows:| Function | Domain | Range |
|---|---|---|
| arcsinh(x) | (−∞, ∞) | (−∞, ∞) |
| arccosh(x) | [1, ∞) | [0, ∞) |
| arctanh(x) | (−1, 1) | (−∞, ∞) |
| arccoth(x) | (−∞, −1] ∪ [1, ∞) | (−∞, ∞) |
| arcsech(x) | (0, 1] | [0, ∞) |
| arccsch(x) | (−∞, 0) ∪ (0, ∞) | (−∞, ∞) |
Historical development
The hyperbolic functions, precursors to their inverses, emerged in the mid-18th century through the work of Italian mathematician Vincenzo Riccati, who introduced them around 1757 as tools for solving cubic equations by analogy to circular functions.[8] Independently, Swiss polymath Johann Heinrich Lambert advanced their study in the 1760s, providing the first systematic exposition in his 1768 paper on transcendental functions, where he established key properties and notation like sinh and cosh.[9] These early developments laid the groundwork for inverse hyperbolic functions, which arose naturally as means to invert these relations, much like inverse trigonometric functions addressed arc lengths on circles. In the 19th century, the explicit development and application of inverse hyperbolic functions gained momentum, particularly for tackling transcendental equations that resisted algebraic solutions. Mathematicians incorporated them into geometric and analytical frameworks, leveraging their utility in projective geometry and equation solving during the 1820s and 1830s. This period also benefited from Leonhard Euler's earlier 18th-century insights into complex analysis, where relations like sinh(iz) = i sin(z) illuminated the logarithmic expressions underlying inverse hyperbolic functions, fostering their integration into broader calculus techniques.[10] The 20th century brought standardization and expanded applications, with inverse hyperbolic functions finding prominence in physics through Hermann Minkowski's 1908 formulation of spacetime in special relativity, where rapidity—the hyperbolic analog of angular velocity—relied on inverses like artanh for velocity compositions.[11] Concurrently, their role in integration techniques solidified, enabling substitutions that simplify definite integrals in engineering and physics, as documented in early 20th-century mathematical tables.[12] Notation was formalized internationally in ISO 80000-2 (2009), recommending prefixes like "ar-" for inverses (e.g., arsinh) to ensure consistency in mathematical communication.Fundamentals
Notation
The inverse hyperbolic functions are denoted in various ways across mathematical literature and standards, reflecting both historical conventions and efforts toward uniformity. Common notations include a prefix such as "arcsinh" or "arsinh" applied to the base hyperbolic function name, as seen in many textbooks and reference works for the inverse hyperbolic sine, with analogous forms for the others (e.g., "arcosh" or "arccoth").[6] Another widespread convention uses a superscript inverse indicator, such as \sinh^{-1} for the inverse hyperbolic sine, which emphasizes the functional inverse relationship and is prevalent in computational and analytical contexts to distinguish it from reciprocals.[13] The International Organization for Standardization (ISO) 80000-2:2019 specifies the "ar-" prefix (e.g., \arsinh x) as the preferred notation for inverse hyperbolic functions, aligning with similar conventions for inverse trigonometric functions while opting for "ar-" to signify "area hyperbolic" rather than the potentially misleading "arc-" that evokes arc length in trigonometric inverses. This standard promotes consistency in scientific and technical documentation, recommending the argument be placed in parentheses immediately following the function symbol without intervening spaces (e.g., \arsinh(x)). Variations persist in published texts, particularly regarding capitalization and length: some older or specialized works use "Arcsinh" with a capital "A" for emphasis, while modern preferences favor the lowercase "arsinh" for brevity and alignment with ISO guidelines.[14] The choice between "arc-" and "ar-" stems from a deliberate distinction, as "arc-" borrows from trigonometric notation but inaccurately suggests geometric arc measures, whereas "ar-" accurately reflects the functions' origins in hyperbolic area interpretations.[15] In programming and computational libraries, notations are often abbreviated without prefixes for efficiency; for instance, the C++ standard library in<cmath> provides the function asinh to compute the inverse hyperbolic sine of a floating-point argument. Similar conventions appear in languages like Python (via math.asinh) and MATLAB, prioritizing short, unambiguous identifiers over full descriptive names.[16]
Pronunciation of these notations typically follows phonetic patterns akin to their trigonometric counterparts, with "arcsinh" rendered as "ark-sine-h" or "ark-shine" in English, and "arsinh" as "ar-sine-h" or "ar-shine," though regional accents may vary the emphasis on syllables.[17]
Logarithmic definitions
The inverse hyperbolic functions admit explicit expressions in terms of the natural logarithm, derived by solving the defining equations y = \sinh x, y = \cosh x, and so forth for x in terms of y. These solutions involve exponentiating to eliminate the hyperbolic functions, resulting in quadratic equations whose roots are expressed using the principal branch of the logarithm to ensure real-valued outputs over the appropriate domains. This logarithmic form highlights the close relationship between hyperbolic and exponential functions, as the hyperbolics themselves are ratios of exponentials.[6] For the inverse hyperbolic sine, \operatorname{arcsinh} x = \ln \left( x + \sqrt{x^2 + 1} \right) holds for all real x \in \mathbb{R}. This arises from solving x = \sinh y = \frac{e^y - e^{-y}}{2}, which rearranges to the quadratic e^{2y} - 2x e^y - 1 = 0 in e^y; the positive root e^y = x + \sqrt{x^2 + 1} yields the logarithmic expression upon taking the natural log. The domain is unrestricted on the reals because \sinh y is bijective from \mathbb{R} to \mathbb{R}.[6] The inverse hyperbolic cosine is given by \operatorname{arccosh} x = \ln \left( x + \sqrt{x^2 - 1} \right) for x \geq 1. It derives from x = \cosh y = \frac{e^y + e^{-y}}{2} with y \geq 0 for the principal branch, leading to e^{2y} - 2x e^y + 1 = 0; the root e^y = x + \sqrt{x^2 - 1} (choosing the greater than 1 solution) gives the formula. The restriction x \geq 1 reflects the range of \cosh y for real y, ensuring a real inverse.[6] For the inverse hyperbolic tangent, \operatorname{arctanh} x = \frac{1}{2} \ln \left( \frac{1 + x}{1 - x} \right) applies when |x| < 1. Solving x = \tanh y = \frac{e^y - e^{-y}}{e^y + e^{-y}} simplifies to e^{2y} = \frac{1 + x}{1 - x}, so y = \frac{1}{2} \ln \left( \frac{1 + x}{1 - x} \right). The domain |x| < 1 corresponds to the range of \tanh y over real y.[6] The inverse hyperbolic cotangent is \operatorname{arccoth} x = \frac{1}{2} \ln \left( \frac{x + 1}{x - 1} \right) for |x| > 1. This follows from the relation \operatorname{arccoth} x = \operatorname{arctanh} (1/x) , substituting into the arctanh formula; it solves x = \coth y = \frac{\cosh y}{\sinh y} with the principal value ensuring the correct branch. The domain excludes |x| \leq 1 due to the range of \coth y.[6] For the inverse hyperbolic secant, \operatorname{arcsech} x = \ln \left( \frac{1 + \sqrt{1 - x^2}}{x} \right) when $0 < x \leq 1. Derived from \operatorname{arcsech} x = \operatorname{arccosh} (1/x) , it addresses x = \sech y = 1/\cosh y for y \geq 0; the logarithmic form ensures non-negativity. The domain restriction matches the range of \sech y over non-negative real y.[6] Finally, the inverse hyperbolic cosecant is \operatorname{arccsch} x = \ln \left( \frac{1}{x} + \sqrt{\frac{1}{x^2} + 1} \right) for all real x \neq 0. This comes from \operatorname{arccsch} x = \operatorname{arcsinh} (1/x) , solving x = \csch y = 1/\sinh y; the expression handles both positive and negative x via the principal branch, with the singularity at zero reflecting the undefined \csch 0.[6]Identities
Addition formulae
The addition formulae for inverse hyperbolic functions express the sum or difference of two such functions in terms of another inverse hyperbolic function applied to a combination of their arguments. These identities are analogous to those for inverse trigonometric functions but account for the hyperbolic nature, including specific domain restrictions to ensure real-valued results and avoid branch cuts. Unlike the hyperbolic functions themselves, the sums of inverse hyperbolic functions do not generally simplify to the inverse of the sum of arguments; for example, \arsinh(x + y) = \arsinh(x) + \arsinh(y) holds only under restrictive conditions such as xy = 0 or specific sign alignments, but the general case requires the dedicated addition formula below.[18] For the inverse hyperbolic sine, the addition and subtraction formulae are \arsinh u \pm \arsinh v = \arsinh\left( u \sqrt{1 + v^2} \pm v \sqrt{1 + u^2} \right), where u, v \in \mathbb{R}, and the principal branch is taken with the positive square roots; the equation holds provided the right-hand side argument lies in the domain of \arsinh, which is all reals. To derive this, let \alpha = \arsinh u and \beta = \arsinh v, so u = \sinh \alpha and v = \sinh \beta. Then \arsinh u + \arsinh v = \alpha + \beta, and applying the hyperbolic sine addition formula gives \sinh(\alpha + \beta) = \sinh \alpha \cosh \beta + \cosh \alpha \sinh \beta = u \sqrt{1 + v^2} + v \sqrt{1 + u^2}, since \cosh(\arsinh z) = \sqrt{1 + z^2} for z \in \mathbb{R}. Taking \arsinh of both sides yields the identity; the subtraction case follows similarly using \sinh(\alpha - \beta) = \sinh \alpha \cosh \beta - \cosh \alpha \sinh \beta, with appropriate branch choices to maintain the principal value.[18] For the inverse hyperbolic cosine, defined for arguments at least 1 in the reals, the formulae are \arccosh u \pm \arccosh v = \arccosh\left( u v \pm \sqrt{(u^2 - 1)(v^2 - 1)} \right), valid for u, v \geq 1, ensuring the right-hand side argument is at least 1 and the principal (non-negative) branch is selected. The derivation proceeds analogously: let \alpha = \arccosh u and \beta = \arccosh v, so u = \cosh \alpha and v = \cosh \beta with \alpha, \beta \geq 0. Then \cosh(\alpha + \beta) = \cosh \alpha \cosh \beta + \sinh \alpha \sinh \beta = u v + \sqrt{(u^2 - 1)(v^2 - 1)}, using \sinh(\arccosh z) = \sqrt{z^2 - 1} for z \geq 1; applying \arccosh gives the sum formula, and the difference uses \cosh(\alpha - \beta) = \cosh \alpha \cosh \beta - \sinh \alpha \sinh \beta. These hold without branch issues in the specified domain, as the expressions remain non-negative.[18] The inverse hyperbolic tangent has simpler logarithmic-based formulae: \arctanh u \pm \arctanh v = \arctanh\left( \frac{u \pm v}{1 \pm u v} \right), for |u| < 1, |v| < 1, to keep the argument in (-1, 1) and ensure the sum or difference is real-valued within the principal branch. This can be derived from the logarithmic definition \arctanh z = \frac{1}{2} \ln \left( \frac{1 + z}{1 - z} \right): the sum is \frac{1}{2} \ln \left( \frac{1 + u}{1 - u} \right) + \frac{1}{2} \ln \left( \frac{1 + v}{1 - v} \right) = \frac{1}{2} \ln \left( \frac{(1 + u)(1 + v)}{(1 - u)(1 - v)} \right) = \frac{1}{2} \ln \left( \frac{1 + u + v + u v}{1 - u - v + u v} \right). Simplifying the argument yields \frac{u + v}{1 + u v}, confirming the identity; the subtraction case follows by replacing v with -v, with the conditions preventing poles in the logarithm.[18] Mixed addition formulae also exist, such as for the inverse hyperbolic sine and cosine: \arsinh u \pm \arccosh v = \arsinh\left( u v \pm \sqrt{(1 + u^2)(v^2 - 1)} \right) = \arccosh\left( v \sqrt{1 + u^2} \pm u \sqrt{v^2 - 1} \right), valid for u \in \mathbb{R}, v \geq 1, with domains ensuring the right-hand side arguments are in the respective function domains and principal branches are maintained. The derivation follows similarly by applying the hyperbolic addition formulae and inverting.[18] For the inverse hyperbolic tangent and cotangent: \arctanh u \pm \arccoth v = \arctanh\left( \frac{u v \pm 1}{v \pm u} \right), for |u| < 1, |v| > 1, ensuring the argument lies in (-1, 1). Symmetrically, \arccoth u \pm \arctanh v = \arccoth\left( \frac{u \pm v}{u v \pm 1} \right), for |u| > 1, |v| < 1, keeping the argument outside [-1, 1]. These can be derived using the hyperbolic identities for \tanh and \coth. Addition formulae for \arccoth \pm \arccoth follow analogously to \arccosh \pm \arccosh, replacing hyperbolic cosine with cotangent relations. Formulae for \arcsech and \arccsch can be obtained via the reciprocal identities below.[18]Other identities
The reciprocal relations among inverse hyperbolic functions express the less common functions in terms of the primary ones. Specifically, the inverse hyperbolic cosecant, secant, and cotangent are given by \operatorname{arccsch} z = \operatorname{arcsinh} \left( \frac{1}{z} \right), \operatorname{arcsech} z = \operatorname{arccosh} \left( \frac{1}{z} \right), \operatorname{arccoth} z = \operatorname{arctanh} \left( \frac{1}{z} \right), for appropriate domains where the expressions are defined, such as |z| > 1 for the principal real branches of \arccoth. Multiple-angle formulas for inverse hyperbolic functions can be derived from the addition formulae. The double-angle formula for the inverse hyperbolic tangent is $2 \operatorname{arctanh} x = \operatorname{arctanh} \left( \frac{2x}{1 + x^2} \right), \quad |x| < 1. This holds for the principal real value. Series expansions provide insight into inequalities for these functions near zero. The Maclaurin series for the inverse hyperbolic sine is \operatorname{arcsinh} x = x - \frac{1}{6} x^3 + \frac{3}{40} x^5 - \frac{5}{112} x^7 + \cdots, \quad x \in \mathbb{R}, implying \operatorname{arcsinh} x < x for $0 < x < 1.[13] Similarly, the series for the inverse hyperbolic tangent is \operatorname{arctanh} x = x + \frac{1}{3} x^3 + \frac{1}{5} x^5 + \frac{1}{7} x^7 + \cdots, \quad |x| < 1, yielding \operatorname{arctanh} x > x for $0 < x < 1.[19] These inequalities follow from the signs of the higher-order terms in the expansions.Relations and Compositions
Conversions between inverse hyperbolic functions
The inverse hyperbolic functions for reciprocal hyperbolic functions exhibit straightforward conversion formulas due to the reciprocal relationships among the hyperbolic functions themselves, such as \coth y = 1/\tanh y and \sech y = 1/\cosh y. These conversions allow expressing one inverse function in terms of another by taking reciprocals of the arguments and adjusting for the appropriate branches and domains.[4] A key relation is between the inverse hyperbolic tangent and cotangent: \arccoth x = \arctanh(1/x) for |x| > 1. This follows from the logarithmic definitions, where \arctanh z = \frac{1}{2} \ln \left( \frac{1+z}{1-z} \right) for |z| < 1 and \arccoth z = \frac{1}{2} \ln \left( \frac{z+1}{z-1} \right) for |z| > 1, with substitution yielding the identity; equivalently, \arctanh x = \arccoth(1/x) for $0 < x < 1. The principal branch ensures the result is real and positive for positive arguments outside the unit interval.[4] Similarly, for the secant and cosecant inverses, \arcsech x = \arccosh(1/x) holds for $0 < x \leq 1. Here, if y = \arcsech x, then \sech y = x, so \cosh y = 1/x and y = \arccosh(1/x), with the principal value taken nonnegative. The domain restriction ensures $1/x \geq 1, aligning with the domain of \arccosh.[20] For the cosecant inverse, \arccsch x = \arcsinh(1/x) for x \neq 0, but sign adjustments are necessary to match principal branches: the result has the same sign as x, since \arcsinh is odd and \csch y = 1/\sinh y preserves the sign of y. For x > 0, y > 0; for x < 0, y < 0. This relation stems from solving \csch y = x to get \sinh y = 1/x.[20][4] These patterns generalize from the reciprocal definitions of the hyperbolic functions, enabling efficient computation or simplification in applications like integration and complex analysis, while respecting branch cuts (e.g., for \arccoth, along [-1, 1]) to maintain analytic continuation.[18]Compositions with hyperbolic functions
The compositions of hyperbolic functions with their corresponding inverse hyperbolic functions yield both trivial and non-trivial identities, which follow directly from the definitions and fundamental hyperbolic relations. For the hyperbolic sine and its inverse, the trivial identity holds: \sinh(\arsinh x) = x for all real x \in \mathbb{R}. Similarly, for the hyperbolic cosine, \cosh(\arcosh x) = x for x \geq 1. These identities reflect the inverse nature of the functions within their principal domains on the real line.[21] Non-trivial compositions arise when pairing a hyperbolic function with the inverse of its "adjacent" counterpart, analogous to the geometric interpretations on the hyperbola. Specifically, \cosh(\arsinh x) = \sqrt{1 + x^2} for all real x \in \mathbb{R}, derived from the identity \cosh^2 y - \sinh^2 y = 1 by setting y = \arsinh x. Likewise, \sinh(\arcosh x) = \sqrt{x^2 - 1} for x \geq 1, ensuring the positive branch aligns with the range of \arcosh. These expressions provide algebraic simplifications useful in calculus and physics applications.[22] For the remaining pairs, the trivial identities extend analogously: \tanh(\arctanh x) = x for |x| < 1, and \sech(\arcsech x) = x for $0 < x \leq 1. The hyperbolic cotangent follows \coth(\arccoth x) = x for |x| > 1, and the cosecant \csch(\arccsch x) = x for x \neq 0. These hold under the principal real branches, where the inverses are defined to output real values within specified intervals.[21] In general, these compositions embody the hyperbolic Pythagorean theorem, where the "adjacent" function (e.g., cosh to sinh) yields the hypotenuse-like term \sqrt{1 + (\sinh y)^2} or similar, preserving the hyperbola's metric. While primarily defined for real arguments as above, the identities extend to complex numbers via analytic continuation, with principal values respecting branch cuts in the complex plane.[22]Compositions with trigonometric functions
The compositions of inverse hyperbolic functions with trigonometric or inverse trigonometric functions often involve imaginary units, reflecting the deep connection between hyperbolic and circular functions established through Euler's formula, where hyperbolic functions are trigonometric functions evaluated at imaginary arguments. These relations enable expressions of inverse hyperbolics in terms of inverse trig functions, facilitating computations and extensions in the complex domain. A fundamental identity is \operatorname{arctanh}(i x) = i \arctan x for real x with |x| < 1, where the left side is purely imaginary and the right side scales the real-valued arctangent by i. More generally, for complex z, \operatorname{arctanh} z = i \arctan(-i z), with the principal value defined accordingly. Similarly, \operatorname{arcsinh} x = -i \arcsin(i x) for real x, and \operatorname{arccosh} x = -i \arccos(i x) for real x \geq 1. Analogous forms hold for the other inverse hyperbolic functions, such as \operatorname{arccoth} z = i \arccot(i z). These identities are valid within the principal branches and reveal real-domain limitations: for real inputs, the compositions produce real outputs only when the imaginary scaling aligns the argument within the real range of the inverse trigonometric function, such as for \operatorname{arcsinh} x where \arcsin(i x) is adjusted to yield a real result via the factor of -i. Outside these domains, the results are complex, highlighting the need for careful branch selection. Such compositions support analytic continuation of inverse hyperbolic functions across the complex plane by leveraging the well-studied branch structures of inverse trigonometric functions. They also prove valuable in solving transcendental equations that mix trigonometric and hyperbolic terms, as encountered in models from physics and engineering, where unified complex representations simplify resolutions.Calculus
Derivatives
The first derivatives of the inverse hyperbolic functions can be obtained by differentiating their logarithmic definitions, as given in the logarithmic definitions section, using the chain rule or implicit differentiation.[18] For the inverse hyperbolic sine, the derivative is \frac{d}{dx} \arsinh x = \frac{1}{\sqrt{1 + x^2}}, \quad x \in \mathbb{R}. This holds for all real arguments due to the entire real line domain of \arsinh x.[18] The derivative of the inverse hyperbolic cosine is \frac{d}{dx} \arccosh x = \frac{1}{\sqrt{x^2 - 1}}, \quad x > 1. The principal branch requires x \geq 1, but the derivative is defined for x > 1 to avoid the singularity at the branch point.[18] For the inverse hyperbolic tangent, \frac{d}{dx} \arctanh x = \frac{1}{1 - x^2}, \quad |x| < 1. This reflects the domain restriction to prevent poles at x = \pm 1.[18] The inverse hyperbolic cotangent has the derivative \frac{d}{dx} \arccoth x = \frac{1}{1 - x^2}, \quad |x| > 1. The form is identical to that of \arctanh x, but the domain excludes the interval (-1, 1) where the function is not real-valued in the principal branch.[18] For the inverse hyperbolic secant, \frac{d}{dx} \arcsech x = -\frac{1}{x \sqrt{1 - x^2}}, \quad 0 < x < 1. The negative sign arises from the decreasing nature of the function over its principal domain, and the expression is undefined at x = 0 and x = 1.[18] Finally, the derivative of the inverse hyperbolic cosecant is \frac{d}{dx} \arccsch x = -\frac{1}{|x| \sqrt{1 + x^2}}, \quad x \neq 0. The absolute value in the denominator ensures the derivative is positive for x > 0 and negative for x < 0, consistent with the odd function property, while the domain excludes zero due to the pole.[18]Series expansions
The Taylor series expansion of the inverse hyperbolic sine function around z = 0 is given by \operatorname{arcsinh} z = \sum_{n=0}^{\infty} (-1)^n \frac{\binom{2n}{n}}{4^n (2n+1)} z^{2n+1} = z - \frac{1}{6} z^3 + \frac{3}{40} z^5 - \frac{5}{112} z^7 + \cdots, which converges for all finite z since \operatorname{arcsinh} z is an entire function. This series can be derived by integrating the binomial expansion of (1 + z^2)^{-1/2}, which is the derivative of \operatorname{arcsinh} z. The Taylor series for the inverse hyperbolic tangent around z = 0 takes the form \operatorname{arctanh} z = \sum_{n=0}^{\infty} \frac{z^{2n+1}}{2n+1} = z + \frac{1}{3} z^3 + \frac{1}{5} z^5 + \frac{1}{7} z^7 + \cdots, converging for |z| < 1 (and on the boundary except at z = \pm 1). It arises from the geometric series expansion of \frac{1}{1 - z^2}, integrated term by term to obtain the antiderivative. For the inverse hyperbolic cosine, which is not analytic at z = 0, the asymptotic Laurent series expansion for large |z| (with |z| > 1) is \operatorname{arccosh} z = \ln(2z) - \sum_{n=1}^{\infty} (-1)^{n+1} \frac{\binom{2n-2}{n-1}}{n (2z)^{2n}} = \ln(2z) - \frac{1}{4z^2} - \frac{3}{32z^4} - \frac{5}{128z^6} - \cdots. This expansion follows from the logarithmic definition of \operatorname{arccosh} z and the binomial series for (2z)^{-2n}. Near the branch point z = 1, a Puiseux series expansion applies: \operatorname{arccosh} z = \sqrt{2(z-1)} \left( 1 + \sum_{n=1}^{\infty} (-1)^n \frac{(2n-1)!!}{2^{2n} n! (2n+1)} (z-1)^n \right), converging for \Re z > 0 and |z-1| < 2. The inverse hyperbolic cotangent has a pole at z = 0, so its expansion around infinity (Laurent series in $1/z) for |z| > 1 is \operatorname{arccoth} z = \sum_{n=1}^{\infty} \frac{z^{-(2n-1)}}{2n-1} = \frac{1}{z} + \frac{1}{3z^3} + \frac{1}{5z^5} + \frac{1}{7z^7} + \cdots. [23] This can be obtained by term-by-term integration of the geometric series for \frac{1}{1 - 1/z^2}, the derivative of \operatorname{arccoth} z.Indefinite integrals
Inverse hyperbolic functions frequently appear as antiderivatives in the evaluation of indefinite integrals involving square roots of quadratic expressions. These integrals arise naturally in calculus and are derived from the logarithmic definitions of the inverse hyperbolic functions, which facilitate computation without special functions.[24] A fundamental example is the integral \int \frac{dx}{\sqrt{x^2 + a^2}} = \sinh^{-1}\left(\frac{x}{a}\right) + C, \quad a > 0. This can be verified by differentiation and is equivalent to the logarithmic form \int \frac{dx}{\sqrt{x^2 + a^2}} = \ln\left(x + \sqrt{x^2 + a^2}\right) + C, which is often preferred for numerical evaluation.[24][25] Similarly, for the inverse hyperbolic cosine, \int \frac{dx}{\sqrt{x^2 - a^2}} = \cosh^{-1}\left(\frac{x}{a}\right) + C, \quad x > a > 0, with the logarithmic equivalent \int \frac{dx}{\sqrt{x^2 - a^2}} = \ln\left(x + \sqrt{x^2 - a^2}\right) + C. The domain restriction ensures the expression under the square root is positive.[24][25] For the inverse hyperbolic tangent, \int \frac{dx}{a^2 - x^2} = \frac{1}{a} \tanh^{-1}\left(\frac{x}{a}\right) + C, \quad |x| < a, which corresponds to the logarithmic form \int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln\left|\frac{a + x}{a - x}\right| + C. This integral converges within the specified interval to avoid singularities.[24][25] Analogous forms exist for the remaining inverse hyperbolic functions. For instance, \int \frac{dx}{x \sqrt{x^2 + 1}} = -\csch^{-1}(x) + C, \quad x > 0, derived from the derivative of the inverse hyperbolic cosecant and expressible logarithmically as \int \frac{dx}{x \sqrt{x^2 + 1}} = -\ln\left(\frac{1 + \sqrt{1 + x^2}}{x}\right) + C. These can be confirmed by differentiation of the inverse functions.[26][6] Such integrals have played a central role in mathematical tables since the mid-20th century, notably in Gradshteyn and Ryzhik's comprehensive compilation, first published in Russian in 1948 and widely used for reference in advanced computations involving hyperbolic expressions.[27]Complex Analysis
Principal value of the inverse hyperbolic sine
The principal value of the inverse hyperbolic sine function, denoted \operatorname{arcsinh} z, is defined for a complex number z by the formula \operatorname{arcsinh} z = \ln \left( z + \sqrt{z^2 + 1} \right), where the principal branch of the complex logarithm is used, with its argument restricted to the interval (-\pi, \pi], and the principal branch of the square root is employed such that \sqrt{z^2 + 1} has nonnegative real part when z is real and nonnegative.[28] This expression provides the principal value, which is analytic throughout the complex plane except along specified branch cuts. The branch cuts for \operatorname{arcsinh} z are conventionally placed along the segments of the imaginary axis from -i\infty to -i and from i to i\infty, with branch points at z = i and z = -i.[6] These cuts arise from the multi-valued nature of the logarithm and square root in the defining expression, rendering the unrestricted inverse hyperbolic sine multi-valued, with branches differing by integer multiples of i\pi.[4] The principal branch selects a single analytic continuation from the real axis, where \operatorname{arcsinh} x is real for real x, into the cut plane. At the branch points, the principal value evaluates to \operatorname{arcsinh} i = i \pi / 2 and \operatorname{arcsinh} (-i) = -i \pi / 2, consistent with the principal logarithm applied to the limiting form of the expression.[28] Across the branch cuts, the function exhibits discontinuities: for z = iy with y > 1, the values on either side of the cut are \ln \left( y + \sqrt{y^2 - 1} \right) + i \pi / 2 and \ln \left( y + \sqrt{y^2 - 1} \right) - i \pi / 2, differing by i\pi; a similar jump of -i\pi occurs for y < -1.[29] These discontinuities ensure the principal branch remains single-valued and analytic in the domain \mathbb{C} \setminus ((-\infty, -i] \cup [i, \infty)) i.Principal value of the inverse hyperbolic cosine
The principal value of the inverse hyperbolic cosine function in the complex plane, denoted \arccosh z, is defined by \arccosh z = \ln\left(z + \sqrt{z^2 - 1}\right), where \ln denotes the principal branch of the complex logarithm (with imaginary part in (-\pi, \pi]) and \sqrt{\cdot} denotes the principal branch of the square root (with nonnegative real part). This logarithmic expression selects a specific branch of the multivalued inverse, rendering the function non-entire due to the inherent branch points at z = \pm 1 and the associated discontinuities.[6] The principal branch features a branch cut along the ray (-\infty, 1] on the real axis, across which the function is discontinuous, ensuring analyticity in the cut plane \mathbb{C} \setminus (-\infty, 1]. The range of this principal branch consists of all complex numbers w = u + iv such that u \geq 0 and $0 \leq v \leq \pi.[6] For real arguments z \geq 1, \arccosh z yields a real, nonnegative value, coinciding with the standard real inverse hyperbolic cosine. In contrast, for real z satisfying |z| < 1, \arccosh z is purely imaginary and complex, with the imaginary part lying in (0, \pi).[6] The principal value satisfies the relation \arccosh z = i \arccos( i z ), linking it to the principal branch of the inverse cosine function.[30]Principal values of the inverse hyperbolic tangent and cotangent
The inverse hyperbolic tangent, denoted \operatorname{arctanh} z or \tanh^{-1} z, is defined in the complex plane by the logarithmic expression \operatorname{arctanh} z = \frac{1}{2} \ln \left( \frac{1 + z}{1 - z} \right), where \ln denotes the principal branch of the complex logarithm with branch cut along the negative real axis (-\infty, 0].[31] This definition holds for z \in \mathbb{C} \setminus ((-\infty, -1] \cup [1, \infty)), with branch cuts placed along the real axis segments (-\infty, -1] and [1, \infty) to ensure single-valuedness.[31] The function has branch points at z = \pm 1.[6] The principal branch of \operatorname{arctanh} z maps the cut plane to the infinite horizontal strip -\infty < \operatorname{Re} w < \infty, |\operatorname{Im} w| < \pi/2.[4] On the interval (-1, 1) of the real axis, it takes real values ranging from -\infty to \infty, consistent with the real inverse hyperbolic tangent.[31] Across the branch cuts, the function exhibits a jump discontinuity of i\pi: approaching the cut from above and below differs by i\pi in the value, arising from the $2\pi i jump in the principal logarithm multiplied by the factor of $1/2.[4] The inverse hyperbolic cotangent, denoted \operatorname{arccoth} z or \coth^{-1} z, is defined analogously by \operatorname{arccoth} z = \frac{1}{2} \ln \left( \frac{z + 1}{z - 1} \right), again using the principal logarithm, valid for z \in \mathbb{C} \setminus (-1, 1), with the branch cut along the real interval (-1, 1).[31] Like \operatorname{arctanh} z, it has branch points at z = \pm 1.[6] The functions are reciprocally related by \operatorname{arccoth} z = \operatorname{arctanh}(1/z) for z \neq 0, \pm 1, which interchanges their domains and cuts while preserving the odd nature of both (i.e., \operatorname{arctanh}(-z) = -\operatorname{arctanh} z and similarly for \operatorname{arccoth}).[31] The principal branch of \operatorname{arccoth} z maps the cut plane to the horizontal strip -\infty < \operatorname{Re} w < \infty, |\operatorname{Im} w| < \pi/2.[32] For real z with |z| > 1, it yields real values, positive for z > 1 and negative for z < -1.[31] Near the branch cut (-1, 1), \operatorname{arccoth} z also jumps by i\pi when crossed, due to the same logarithmic mechanism, ensuring the principal value aligns with the reciprocal relation to \operatorname{arctanh}.[4]Principal values of the inverse hyperbolic secant and cosecant
The inverse hyperbolic secant, denoted \operatorname{arcsech} z or \operatorname{sech}^{-1} z, is defined for complex z \neq 0 as the principal value of \operatorname{arccosh}(1/z), where \operatorname{arccosh} refers to the principal branch of the inverse hyperbolic cosine.[6] This relation arises because \operatorname{sech} w = 1 / \operatorname{cosh} w, so inverting yields w = \operatorname{arccosh}(1 / \operatorname{sech} w). The branch structure of \operatorname{arcsech} z inherits from the principal \operatorname{arccosh}, whose branch cut lies along the real interval (-\infty, 1]. Under the mapping u = 1/z, this cut maps to the segments (-\infty, 0] \cup [1, \infty) on the real axis in the z-plane, defining the branch cuts for the principal \operatorname{arcsech}.[33][34] The principal value of \operatorname{arcsech} z admits the explicit logarithmic expression \operatorname{arcsech} z = \ln \left( \frac{1 + \sqrt{1 - z^2}}{z} \right), where the principal branches of the square root (with branch cut along the negative real axis) and logarithm (with argument in (-\pi, \pi]) are employed.[33] An equivalent form is \operatorname{arcsech} z = \ln \left( \frac{1}{z} + \sqrt{\frac{1}{z^2} - 1} \right). The principal range of \operatorname{arcsech} z is the infinite strip \{ w \in \mathbb{C} \mid \operatorname{Re} w \geq 0, \, |\operatorname{Im} w| \leq \pi \}, consistent with the range of the principal \operatorname{arccosh}.[6] Due to the reciprocal mapping $1/z, the function exhibits a pole at z = 0 and discontinuities across the specified real-axis cuts, distinguishing it from inverses like \operatorname{arctanh} that lack cuts on the negative reals.[33] The inverse hyperbolic cosecant, denoted \operatorname{arccsch} z or \operatorname{csch}^{-1} z, is analogously defined as the principal value of \operatorname{arcsinh}(1/z), reflecting the identity \operatorname{csch} w = 1 / \operatorname{sinh} w.[6] The principal \operatorname{arcsinh} has branch cuts along the imaginary segments ( -i \infty, -i ] \cup [ i, i \infty ). The preimage under u = 1/z transforms these to the segments [-i, 0) \cup (0, i] along the imaginary axis in the z-plane, serving as the branch cuts for principal \operatorname{arccsch}.[34] As with \operatorname{arcsech}, a pole occurs at z = 0. The principal value takes the form \operatorname{arccsch} z = \ln \left( \frac{1}{z} + \sqrt{\frac{1}{z^2} + 1} \right), employing the principal square root and logarithm.[35] The principal range is the horizontal strip \{ w \in \mathbb{C} \mid |\operatorname{Im} w| \leq \pi/2, \, \operatorname{Re} w \in \mathbb{R} \}, mirroring that of principal \operatorname{arcsinh}.[6] The reciprocal nature shifts the cuts to the imaginary axis, contrasting with the real-axis cuts for \operatorname{arctanh} and \operatorname{arccoth}, and emphasizes the role of the mapping in adapting the branch structure from the direct inverses.[36]Graphical representation
The graphical representation of inverse hyperbolic functions in the complex plane reveals their multi-valued nature through principal branches, branch cuts, and associated Riemann surfaces. Contour plots of the real and imaginary parts of the principal branch of arcsinh(z) illustrate its analytic continuation across the cut plane, with branch points at z = ±i and cuts along the imaginary axis from -i∞ to -i and from i to i∞. These plots show smooth variations in the real part along the real axis, where arcsinh(z) is real-valued, transitioning to complex values off the axis, highlighting the function's entire-like behavior in the principal domain despite the cuts.[6] Branch cut diagrams for other inverse hyperbolic functions depict the discontinuities in the complex plane. For arccosh(z), the principal branch features a cut along the real axis from -∞ to 1, with branch points at z = ±1, separating regions where the function takes real values for z ≥ 1 from complex values elsewhere. Similarly, arctanh(z) has cuts along the real axis from -∞ to -1 and from 1 to ∞, again with branch points at ±1, confining real values to the open interval (-1, 1) on the real axis. These diagrams often use lines or shaded regions to indicate the cuts, emphasizing how crossing them jumps between sheet values.[6] Riemann surfaces provide a global visualization by stacking sheets to resolve the multi-valuedness, typically forming two-sheeted coverings for inverse hyperbolic functions like arccosh(z), arctanh(z), arccsch(z), arcsech(z), and arccoth(z), except for arcsinh(z) which shares the same two-sheeted structure due to its logarithmic form involving a square root. These surfaces exhibit monodromy where encircling a branch point swaps the sheets, as seen in plots of the imaginary part over the complex plane or Riemann sphere, faithfully representing the topology. For arcsinh(z), the surface connects across the imaginary cuts, showing consistent sheet transitions around the branch points at ±i.[37][38][39] Color-coded domain plots distinguish regions where the principal values are real from those where they are complex, aiding in understanding the functions' behavior. For instance, the real axis segments outside the cuts appear in blue or green for real-valued domains, while the cut planes and off-axis regions are shaded in red or yellow for complex values, with transitions marked by contour lines of constant real or imaginary parts. Such visualizations reference the principal values defined for each function, illustrating their analytic domains without rederiving the branches.[6] Software tools like Mathematica's ComplexPlot and ComplexContourPlot facilitate these representations, generating density plots of real and imaginary components over the complex plane, as well as 3D renderings of Riemann surfaces projected onto the sphere to capture monodromy effects. These tools allow interactive exploration, such as zooming near branch points to observe sheet crossings.Applications
In integration and differential equations
Inverse hyperbolic functions frequently appear as antiderivatives in standard integration problems, providing closed-form solutions to integrals that otherwise require logarithmic expressions. For instance, the integral \int \frac{du}{\sqrt{u^2 + a^2}} = \sinh^{-1}\left(\frac{u}{a}\right) + C for a > 0 evaluates directly using the inverse hyperbolic sine function, derived from the derivative formula \frac{d}{du} [\sinh^{-1}(u/a)] = \frac{1}{a \sqrt{(u/a)^2 + 1}}.[40] Similar forms hold for other inverse hyperbolic functions, such as \int \frac{du}{\sqrt{u^2 - a^2}} = \cosh^{-1}\left(\frac{u}{a}\right) + C for u > a > 0.[40] These integrals are obtained through substitution or recognition of the derivative patterns, offering a hyperbolic alternative to trigonometric substitutions in calculus.[41] In differential equations, inverse hyperbolic functions emerge naturally when solving separable first-order equations, particularly those modeling rates of change in physical systems. Consider the equation y' = \sqrt{1 + y^2}; separating variables yields \int \frac{dy}{\sqrt{1 + y^2}} = \int dx, and recognizing the derivative form, the solution is y = \sinh(x + C).[24] To satisfy an initial condition y(0) = y_0, apply the inverse: x = \sinh^{-1}(y) - \sinh^{-1}(y_0), inverting the hyperbolic sine to express the independent variable explicitly.[42] This approach is common in mechanics for equations describing motion under specific force laws, where the inverse provides the time or position as a function of displacement. These functions also facilitate solutions in physics problems involving variable forces or curves of equilibrium. For example, in calculating the work done against a force F(x) = \frac{k}{\sqrt{x^2 + a^2}}, the work integral W = \int F(x) \, dx = k \sinh^{-1}\left(\frac{x}{a}\right) + C yields the inverse hyperbolic sine directly, modeling scenarios like gravitational or electrostatic potentials in non-uniform fields.[41] In catenary problems, where a hanging chain forms the curve y = a \cosh(x/a), determining the parameter a from measured sag or length often requires inverting hyperbolic relations; for instance, the arc length s = a \sinh(x/a) requires numerically solving the transcendental equation s = a \sinh(x/a) to fit experimental data.[24] Such applications highlight the analytical utility of inverse hyperbolic functions in approximating solutions to more complex integrals, including reductions for elliptic integrals in advanced contexts like pendulum motion analogs.[42]In hyperbolic geometry and special relativity
In hyperbolic geometry, inverse hyperbolic functions play a central role in measuring distances within various models of non-Euclidean space. In the hyperboloid model, points lie on the upper sheet of the hyperboloid defined by x_1^2 + \cdots + x_n^2 - x_{n+1}^2 = -1 with x_{n+1} > 0 in Minkowski space \mathbb{R}^{n,1}. The hyperbolic distance d(P, Q) between two points P and Q on this hyperboloid is given by d(P, Q) = \cosh^{-1}(-\langle P, Q \rangle), where \langle \cdot, \cdot \rangle denotes the Minkowski inner product; this formula arises from parameterizing geodesics as intersections of the hyperboloid with 2-planes through the origin and integrating the induced Riemannian metric along these paths.[43] Similarly, in the Poincaré disk model, which represents hyperbolic space as the open unit disk with a conformal metric, the distance between points z, w \in \mathbb{D} is d_h(z, w) = \arcosh\left(1 + 2 \frac{|z - w|^2}{(1 - |z|^2)(1 - |w|^2)}\right);[44] In special relativity, inverse hyperbolic functions parameterize Lorentz boosts and velocities in a way that simplifies additions and transformations, drawing analogies to rotations via hyperbolic geometry. The rapidity \phi, defined as \phi = \artanh(v/c) where v is the relative velocity and c the speed of light, ranges over all real numbers while v is bounded by c, ensuring additive composition under successive boosts: if two boosts have rapidities \phi_1 and \phi_2, the total is \phi = \phi_1 + \phi_2.[45] The Lorentz factor \gamma = 1 / \sqrt{1 - v^2/c^2} corresponds to \gamma = \cosh \phi, with \beta \gamma = \sinh \phi where \beta = v/c, allowing the Lorentz transformation to be expressed as a hyperbolic rotation matrix.[46] In Minkowski spacetime, this parametrization describes the worldline of a uniformly moving particle: for a boost along the z-direction, the coordinates transform as ct = (ct') \cosh \phi + z' \sinh \phi and z = (ct') \sinh \phi + z' \cosh \phi, or inversely, the position of a particle with proper time \tau and rapidity \phi is x = c \tau \cosh \phi, ct = c \tau \sinh \phi in the observer's frame (assuming motion along x).[47] These functions also appear in general relativity, particularly in coordinate systems for black hole metrics. In Kruskal-Szekeres coordinates for the Schwarzschild black hole, which extend the spacetime across the event horizon to reveal its maximal analytic structure, the transformation from Schwarzschild time t and radial coordinate r involves hyperbolic functions: specifically, \tanh(t/4M) = V/U outside the horizon (where M is the black hole mass, and U, V are Kruskal null coordinates), implying t = 4M \artanh(V/U); this ensures regularity at r = 2M and covers regions including the black hole interior and white hole.[48][49]Numerical evaluation
Inverse hyperbolic functions are typically computed using their explicit logarithmic expressions, which provide a direct and efficient method for numerical evaluation in most cases. For real arguments within the appropriate domains, the formulas are \operatorname{asinh}(x) = \ln\left(x + \sqrt{x^2 + 1}\right) for all real x, \operatorname{acosh}(x) = \ln\left(x + \sqrt{x^2 - 1}\right) for x \geq 1, and \operatorname{atanh}(x) = \frac{1}{2} \ln\left(\frac{1 + x}{1 - x}\right) for |x| < 1, with analogous forms for the remaining functions derived from these.[6] These expressions require careful implementation to handle branch cuts in the complex plane, where principal values are selected by choosing the argument of the logarithm such that its imaginary part lies in (-\pi, \pi], ensuring continuity across the cuts (e.g., along the imaginary axis for \operatorname{asinh}).[6] In libraries, functions likefplog (floating-point logarithm) are employed to manage precision and overflow during evaluation.[50]
For small arguments where |x| \ll 1, direct use of the logarithmic forms can lead to underflow or loss of precision, so Taylor series expansions are preferred for accuracy. For instance, \operatorname{atanh}(x) \approx x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots converges rapidly in this regime, often truncated after a few terms with prestored coefficients to achieve up to 25 decimal digits of precision.[50] Similar polynomial or rational approximations (e.g., x + x^3 P(x)/Q(x) for \operatorname{atanh}(x) when |x| < 0.5) are used in implementations like the Cephes library, which underpins many modern systems.[51]
For large arguments, asymptotic approximations simplify computation and prevent overflow in the square root terms. Specifically, \operatorname{asinh}(x) \approx \operatorname{sgn}(x) \ln(2 |x|) as |x| \to \infty, while \operatorname{acosh}(x) \approx \ln(2x) for x \to \infty, reducing the evaluation to a logarithm after scaling.[50] These are applied when |x| > 10^{16} in double precision to maintain stability, with the full logarithmic form used otherwise after argument reduction.[50]
Special functions libraries implement these methods with safeguards against overflow and underflow. The GNU Scientific Library (GSL) provides routines like gsl_asinh and gsl_acosh, using logarithmic formulas with internal checks for domain validity and precision limits tied to machine epsilon (e.g., returning values bounded by \ln(2 \Omega), where \Omega is the overflow threshold).[52] Similarly, SciPy's scipy.special.asinh (and equivalents) relies on NumPy's vectorized ufuncs, which employ the logarithmic form with fused operations to handle large inputs via asymptotic scaling and small inputs via series, ensuring relative errors below machine precision (e.g., 0.5 ulps for \operatorname{acosh}(x) \geq 1.21).[50] Overflow is managed by capping outputs at approximately \ln(2 \times 10^{308}) for double precision, while underflow near zero uses series to avoid subtracting near-unity terms in logarithms.[50]
Key challenges in numerical evaluation include subtractive cancellation in the logarithmic arguments, particularly for \operatorname{acosh}(x) near x=1^+, where \sqrt{x^2 - 1} is small and direct computation of x^2 - 1 may lose digits in floating-point arithmetic. To mitigate this, an alternative form \sqrt{2(x-1)} is used for the square root when x is close to 1, preserving accuracy by avoiding the difference of close squares (e.g., computing \sqrt{(x-1)(x+1)} \approx \sqrt{2(x-1)} since x+1 \approx 2).[50] Branch handling for complex arguments further requires selecting the principal logarithm to adhere to the defined cuts, as implemented in libraries supporting complex inputs.[6]