Hyperbolic functions

Hyperbolic functions are a class of mathematical functions analogous to the trigonometric functions, but defined using the exponential function and associated with the geometry of the hyperbola rather than the circle. The primary hyperbolic functions are the hyperbolic sine (sinh x), hyperbolic cosine (cosh x), and hyperbolic tangent (tanh x), with additional functions including hyperbolic secant (sech x), cosecant (csch x), and cotangent (coth x). These functions satisfy identities similar to those of trigonometric functions, such as cosh²x - sinh²x = 1, but they model exponential growth and decay rather than periodic behavior.^[1]^[2] The hyperbolic functions were first developed in the 18th century, with Italian mathematician Vincenzo Riccati (1707–1775) introducing them through the geometry of the unit hyperbola x² - y² = 1 to solve differential equations arising from physical problems. Shortly thereafter, Johann Heinrich Lambert (1728–1777) formalized their trigonometric-like properties and applications, linking them explicitly to exponential expressions. Their definitions are given by sinh x = (e^x - e^-x)/2 and cosh x = (e^x + e^-x)/2, from which tanh x = sinh x / cosh x follows, along with the reciprocal functions.^[3]^[2] Key properties include the even nature of cosh x (where cosh(-x) = cosh x) and the odd nature of sinh x (where sinh(-x) = -sinh x), with tanh x also odd and approaching ±1 asymptotically as x → ±∞. Derivatives follow simple rules, such as d/dx [cosh x] = sinh x and d/dx [sinh x] = cosh x, enabling their use in integration techniques. Addition formulas mirror trigonometric ones, for example, sinh(x + y) = sinh x cosh y + cosh x sinh y. Inverse hyperbolic functions, like arsinh x = ln(x + √(x² + 1)), are defined for appropriate domains and have logarithmic expressions.^[1]^[2] Hyperbolic functions have wide applications in physics and engineering, including modeling the shape of hanging chains (catenaries, described by cosh x), satellite orbits, and wave propagation. In special relativity, they parameterize Lorentz transformations and describe rapidity in velocity addition. Their exponential foundations make them essential for solving differential equations in areas like heat transfer and electrical circuits.^[1]

History and Notation

Historical Development

The study of hyperbolas dates back to ancient Greek geometry, where they were first recognized as one of the conic sections. Around 350 BC, Menaechmus discovered the hyperbola while attempting to solve the Delian problem of duplicating the cube, describing it as the intersection of a cone with a plane. Later, in the 3rd century BC, Apollonius of Perga provided a systematic treatment in his work Conics, naming the curve "hyperbola" (meaning "excess") and developing its properties, including parametric representations that related points on the curve to parameters, laying foundational geometric insights that would later inspire hyperbolic functions.^[4] A key milestone in the evolution toward hyperbolic functions occurred in the late 17th century with the catenary problem, which describes the shape of a hanging chain under gravity. In 1690, Jakob Bernoulli posed this challenge in Acta Eruditorum, and his brother Johann Bernoulli solved it in 1691, deriving the curve's equation through differential methods, though without explicit hyperbolic terminology; the solution's form was later recognized as involving what became the hyperbolic cosine. This application highlighted the utility of such functions in solving differential equations related to physical curves.^[5] The formal introduction of hyperbolic functions emerged in the 18th century. In 1757, Italian mathematician Vincenzo Riccati pioneered their definition in the first volume of Opusculorum ad res physicas et mathematicas pertinentium, expressing sinh and cosh via integrals and linking them to the geometry of the unit hyperbola, complete with addition formulas and derivatives; he denoted them as Sh and Ch. Building on this, Johann Heinrich Lambert provided the first systematic development in his 1761 memoir Mémoire sur les suites, published in 1768, where he defined them logarithmically as "sinus hyperbolicus" and "cosinus hyperbolicus," establishing their trigonometric analogies without complex numbers and popularizing their use in analysis. Leonhard Euler advanced their exponential expressions, such as relating cosh x to (e^x + e^{-x})/2, in works like his 1748 Introductio in analysin infinitorum, refining earlier integral forms into more accessible analytic tools.^[6]^[7]^[5] In the 20th century, a significant milestone came in 1908, when Hermann Minkowski incorporated hyperbolic functions into special relativity, interpreting Lorentz transformations as hyperbolic rotations in spacetime, thus extending their role in physics.^[5]^[8]

Standard Notation

The standard notation for hyperbolic functions employs abbreviations that parallel those of trigonometric functions, with an added "h" to denote the hyperbolic variant. The primary functions are denoted as follows: hyperbolic sine by \sinh x, hyperbolic cosine by \cosh x, hyperbolic tangent by \tanh x, hyperbolic cotangent by \coth x, hyperbolic secant by \sech x, and hyperbolic cosecant by \csch x.^[9]^[10] These symbols, introduced in the 19th century, facilitate consistency with trigonometric notation, where \sin x corresponds to \sinh x and \cos x to \cosh x.^[11] Historical alternatives include \sm x and \cm x for hyperbolic sine and cosine, respectively, as seen in early 20th-century mathematical tables. In physics and older literature, abbreviated forms such as \sh x and \ch x are common for hyperbolic sine and cosine.^[12]^[13] For inverse hyperbolic functions, the notations \sinh^{-1} x, \cosh^{-1} x, \tanh^{-1} x, \coth^{-1} x, \sech^{-1} x, and \csch^{-1} x are standard, with alternative arc notations like \arcsinh x, \arccosh x, \artanh x, and so on also widely used.^[14]^[15] Typographical guidelines recommend rendering these in italicized lowercase letters within mathematical expressions, with the full terms "hyperbolic sine" or "hyperbolic cosine" used in prose for clarity. Pronunciation typically follows "shine" for \sinh x, "cosh" for \cosh x, and "thanch" for \tanh x, though regional variations exist, such as "sinch" or "cynsh" for \sinh x.^[16]

Definitions

Exponential Definitions

The hyperbolic sine and cosine functions are fundamentally defined in terms of the exponential function for real arguments x \in \mathbb{R}. The hyperbolic sine is given by

\sinh x = \frac{e^x - e^{-x}}{2},

while the hyperbolic cosine is

\cosh x = \frac{e^x + e^{-x}}{2}.

These definitions provide a direct means for computation and reveal the functions' close ties to exponential growth and decay.^[17]^[18] The remaining hyperbolic functions are derived from \sinh x and \cosh x. The hyperbolic tangent is the ratio \tanh x = \frac{\sinh x}{\cosh x}, the hyperbolic cotangent is \coth x = \frac{\cosh x}{\sinh x}, the hyperbolic secant is \sech x = \frac{1}{\cosh x}, and the hyperbolic cosecant is \csch x = \frac{1}{\sinh x}. These expressions maintain the real domain and inherit properties from their foundational components.^[19]^[20] A key behavioral aspect stems from the exponential forms: \cosh x is an even function, satisfying \cosh(-x) = \cosh x, whereas \sinh x is odd, with \sinh(-x) = -\sinh x. The derived functions follow suit, with \tanh x, \coth x, and \csch x being odd, and \sech x even. As x \to \infty, \sinh x \to \infty and \cosh x \to \infty, while \tanh x \to 1; symmetrically, as x \to -\infty, \sinh x \to -\infty, \cosh x \to \infty, and \tanh x \to -1. These limits highlight the non-periodic, monotonic nature of the functions for large |x|.^[17]^[21]^[22] The exponential definitions provide a real analog to the trigonometric functions, which are defined via the complex exponential in Euler's formula.

Differential Equation Definitions

Hyperbolic functions can be characterized as solutions to specific linear ordinary differential equations (ODEs). In particular, both the hyperbolic cosine and hyperbolic sine functions serve as fundamental solutions to the second-order linear homogeneous ODE

\frac{d^2 y}{dx^2} - y = 0.

The general solution to this equation is given by

y(x) = A \cosh x + B \sinh x,

where A and B are arbitrary constants determined by initial conditions.^[23]^[21] This form arises because the characteristic equation r^2 - 1 = 0 has roots r = \pm 1, leading to the linear combination of e^x and e^{-x}, though the hyperbolic basis emphasizes the even and odd components.^[24] The individual functions are uniquely specified by their initial conditions at x = 0: \cosh 0 = 1 and \sinh 0 = 0, with derivatives satisfying \frac{d}{dx} \cosh x = \sinh x and \frac{d}{dx} \sinh x = \cosh x. These conditions ensure that \cosh x and \sinh x form a basis for the solution space, guaranteeing uniqueness for the initial value problem.^[23]^[24] Additionally, the hyperbolic sine can be defined as the unique solution to the first-order nonlinear ODE

\frac{dy}{dx} = \sqrt{1 + y^2},

subject to the initial condition y(0) = 0. This follows from the identity \cosh^2 x - \sinh^2 x = 1, which implies \frac{d}{dx} \sinh x = \cosh x = \sqrt{1 + (\sinh x)^2}. Similarly, the hyperbolic cosine satisfies

\frac{dy}{dx} = \sqrt{y^2 - 1}

for |y| \geq 1, with initial condition y(0) = 1; this defines y = \cosh x for x \geq 0 (where \sinh x \geq 0), and the full even function is obtained by extension: \cosh(-x) = \cosh x. These first-order equations highlight the autonomous nature of the functions and their role in separable ODEs.^[23]^[24] Such differential equation formulations connect hyperbolic functions to physical contexts, such as the catenary curve describing a uniformly loaded hanging chain, where the shape y(x) satisfies \frac{d^2 y}{dx^2} = \frac{1}{\sqrt{1 + \left( \frac{dy}{dx} \right)^2}} and resolves to y = a \cosh(x/a).^[23] These definitions via ODEs are equivalent to exponential representations but emphasize analytical solutions to boundary value problems.^[23]

Complex Trigonometric Definitions

Hyperbolic functions can be defined using trigonometric functions evaluated at purely imaginary arguments, providing an analogy between hyperbolic and circular trigonometry in the complex plane. Specifically, the hyperbolic sine and cosine are expressed as

\sinh z = -i \sin(iz),

\cosh z = \cos(iz),

where i is the imaginary unit and \sin and \cos are the standard trigonometric functions.^[10] These definitions establish a direct connection to the exponential forms of the hyperbolic functions. Substituting Euler's formula, e^{i\theta} = \cos \theta + i \sin \theta, into the trigonometric expressions yields the equivalence. For instance, \cos(iz) = \frac{e^{i(iz)} + e^{-i(iz)}}{2} = \frac{e^{-z} + e^{z}}{2} = \cosh z, and similarly, -i \sin(iz) = -i \cdot \frac{e^{i(iz)} - e^{-i(iz)}}{2i} = \frac{e^{z} - e^{-z}}{2} = \sinh z.^[10]^[9] The remaining hyperbolic functions follow analogously from their trigonometric counterparts:

\tanh z = -i \tan(iz), \quad \coth z = i \cot(iz), \quad \sech z = \sec(iz), \quad \csch z = i \csc(iz).

These relations highlight the structural similarities while adapting for the hyperbolic case.^[9]^[25] Unlike the trigonometric functions, which are periodic with period $2\pi on the real line, the hyperbolic functions are non-periodic for real arguments. However, in the complex domain, they exhibit periodicity with imaginary periods, such as $2\pi i for \sinh z and \cosh z, reflecting their trigonometric origins but shifted into the hyperbolic regime.^[10]

Fundamental Identities and Properties

Addition and Subtraction Formulas

The addition and subtraction formulas for hyperbolic functions are fundamental identities that express the hyperbolic sine and cosine of a sum or difference of arguments in terms of products of the individual functions, mirroring the angle-addition theorems in trigonometry but without the alternating signs characteristic of circular functions. These formulas arise naturally from the exponential definitions of the hyperbolic functions and are essential for simplifying expressions involving combined arguments.^[26] The sum formulas are given by:

\sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y

\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y

These can be derived by substituting the exponential definitions of \sinh z = \frac{e^z - e^{-z}}{2} and \cosh z = \frac{e^z + e^{-z}}{2} into z = x + y, expanding, and simplifying the resulting expressions. A similar derivation applies to \cosh(x + y), resulting in the positive product identity.^[26] The difference formulas follow by replacing y with -y in the sum formulas, leveraging the identities \sinh(-y) = -\sinh y and \cosh(-y) = \cosh y:

\sinh(x - y) = \sinh x \cosh y - \cosh x \sinh y

\cosh(x - y) = \cosh x \cosh y - \sinh x \sinh y.

These can be verified analogously via exponential substitution for x - y.^[26] For the hyperbolic tangent, the addition formula is derived by dividing the sum formulas for \sinh and \cosh:

\tanh(x + y) = \frac{\sinh(x + y)}{\cosh(x + y)} = \frac{\sinh x \cosh y + \cosh x \sinh y}{\cosh x \cosh y + \sinh x \sinh y}.

Dividing numerator and denominator by \cosh x \cosh y simplifies this to

\tanh(x + y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y},

which highlights the similarity to the trigonometric tangent addition formula.^[26]

Multiple-Angle and Half-Argument Formulas

Multiple-angle formulas for hyperbolic functions express sinh(nx) and cosh(nx) in terms of powers of sinh(x) and cosh(x), analogous to their trigonometric counterparts but derived from the addition formulas for hyperbolic functions.^[27] The double-angle formulas are fundamental and given by

\sinh(2z) = 2 \sinh z \cosh z

\cosh(2z) = \cosh^2 z + \sinh^2 z = 2 \cosh^2 z - 1 = 1 + 2 \sinh^2 z.

These identities follow directly from applying the addition formula \cosh(a + b) = \cosh a \cosh b + \sinh a \sinh b and \sinh(a + b) = \sinh a \cosh b + \cosh a \sinh b with a = b = z.^[27]^[28] For the triple angle, the formulas are

\sinh(3z) = 3 \sinh z + 4 \sinh^3 z

\cosh(3z) = 4 \cosh^3 z - 3 \cosh z.

These can be obtained by composing the double-angle formulas or using the addition formula iteratively.^[27] Half-argument formulas provide expressions for half-angles in terms of the full argument:

\sinh\left(\frac{z}{2}\right) = \pm \sqrt{\frac{\cosh z - 1}{2}}

\cosh\left(\frac{z}{2}\right) = \pm \sqrt{\frac{\cosh z + 1}{2}}.

The signs depend on the quadrant or branch considered, with the principal values often taken positive for real z \geq 0. These derive from solving the double-angle relations for half-arguments.^[27]

Square and Power Identities

One of the fundamental identities for hyperbolic functions is the hyperbolic Pythagorean theorem, which states that

\cosh^2 x - \sinh^2 x = 1.

This identity is derived directly from the exponential definitions of the hyperbolic functions and serves as an analog to the trigonometric identity \cos^2 x + \sin^2 x = 1. Power-reduction formulas express the squares of hyperbolic sine and cosine in terms of the double-angle hyperbolic cosine:

\sinh^2 x = \frac{\cosh 2x - 1}{2},

\cosh^2 x = \frac{\cosh 2x + 1}{2}.

These formulas follow from rearranging the double-angle identity for \cosh 2x = \cosh^2 x + \sinh^2 x combined with the Pythagorean identity. For higher powers, identities relate cubes and other powers to multiple-angle expressions. For example, the cube of the hyperbolic sine is given by

\sinh^3 x = \frac{\sinh 3x - 3 \sinh x}{4},

which is obtained by solving the triple-angle formula \sinh 3x = 3 \sinh x + 4 \sinh^3 x for the cubic term, yielding $4 \sinh^3 x = \sinh 3x - 3 \sinh x. A similar relation holds for the hyperbolic cosine:

\cosh^3 x = \frac{3 \cosh x + \cosh 3x}{4},

derived from \cosh 3x = 4 \cosh^3 x - 3 \cosh x. These power identities are useful in expanding series representations and solving differential equations involving hyperbolic functions.^[29]

Characterizing Properties of Individual Functions

Hyperbolic Cosine and Sine

The hyperbolic cosine function, denoted \cosh x, and the hyperbolic sine function, denoted \sinh x, are defined in terms of exponentials as \cosh x = \frac{e^x + e^{-x}}{2} and \sinh x = \frac{e^x - e^{-x}}{2}. These functions form the foundational pair of hyperbolic functions, analogous to cosine and sine in trigonometric contexts but exhibiting unbounded growth rather than periodicity.^[30] A key distinguishing property is their symmetry: \cosh x is an even function, satisfying \cosh(-x) = \cosh x for all real x, while \sinh x is an odd function, satisfying \sinh(-x) = -\sinh x. This evenness of \cosh x reflects its symmetry about the y-axis in the real plane, whereas the oddness of \sinh x implies antisymmetry about the origin.^[17] For real arguments, \cosh x \geq 1, achieving its global minimum value of 1 at x = 0, and remains positive everywhere. In contrast, \sinh x passes through the origin with \sinh 0 = 0 and is strictly monotonic increasing over all real x, ranging from -\infty to \infty. These behaviors underscore \cosh x as a convex "U-shaped" curve and \sinh x as a strictly rising S-shaped curve.^[31] As |x| becomes large, both functions exhibit exponential growth: \cosh x \sim \frac{e^{|x|}}{2} and \sinh x \sim \frac{e^{|x|}}{2} \cdot \operatorname{sgn}(x), dominating any polynomial behavior and reflecting their ties to exponential functions. Geometrically, \cosh x generates the shape of a catenary, the curve formed by a uniformly dense hanging chain under gravity, described by y = a \cosh(x/a) for scaling constant a > 0.^[32] The arc length along this catenary from the vertex to a point at parameter x is given by a \sinh(x/a), highlighting \sinh x's role in measuring hyperbolic "distances."^[33]

Hyperbolic Tangent and Cotangent

The hyperbolic tangent function, defined as the ratio \tanh x = \frac{\sinh x}{\cosh x}, and the hyperbolic cotangent function, \coth x = \frac{\cosh x}{\sinh x}, exhibit distinct behaviors arising from their definitions in terms of the fundamental hyperbolic sine and cosine. For real arguments x, the hyperbolic tangent is strictly bounded such that |\tanh x| < 1, with the function being odd and strictly increasing across the entire real line.^[34] As x \to \pm \infty, \tanh x approaches \pm 1 asymptotically, and a more precise approximation for large |x| is \tanh x \sim \operatorname{sign}(x) \left(1 - 2e^{-2|x|}\right).^[35] The addition formula for the hyperbolic tangent is \tanh(u \pm v) = \frac{\tanh u \pm \tanh v}{1 \pm \tanh u \tanh v}, which facilitates computations involving sums or differences of arguments. In contrast, the hyperbolic cotangent has simple poles at x = n\pi i for integers n, including a pole on the real axis at x = 0, rendering it undefined there and introducing discontinuities.^[34] On the real line excluding the origin, \coth x is real-valued and monotonic: strictly decreasing on (0, \infty) from +\infty to +1, and strictly decreasing on (-\infty, 0) from -1 to -\infty.^[34] For large |x|, \coth x \sim \operatorname{sign}(x), approaching \pm 1 as x \to \pm \infty. The addition formula is \coth(u \pm v) = \frac{\coth u \coth v \pm 1}{\coth u \pm \coth v}.

Hyperbolic Secant and Cosecant

The hyperbolic secant function, denoted \operatorname{sech} x, is defined as \operatorname{sech} x = \frac{2}{e^x + e^{-x}}.^[10] It attains a maximum value of 1 at x = 0 and exhibits exponential decay to 0 as |x| increases, approaching 0 asymptotically for large |x|.^[36] This bell-shaped profile makes \operatorname{sech} x particularly useful in modeling localized phenomena, such as optical solitons in nonlinear wave equations, where it describes stable, non-dispersive pulses. The Fourier transform of \operatorname{sech} x is \pi \operatorname{sech}\left(\frac{\pi \xi}{2}\right), highlighting its self-similar transform properties under certain scalings.^[37] The hyperbolic cosecant function, denoted \operatorname{csch} x, is defined as \operatorname{csch} x = \frac{2}{e^x - e^{-x}}.^[10] As an odd function, \operatorname{csch}(-x) = -\operatorname{csch} x, it features a pole at x = 0 where it diverges to \pm \infty.^[36] For large positive x, \operatorname{csch} x decays exponentially to 0, asymptotically \operatorname{csch} x \sim 2 e^{-x}; for large negative x, \operatorname{csch} x \sim -2 e^{x}.^[1] Both functions are integrable over appropriate intervals. The indefinite integral of \operatorname{sech} x is \int \operatorname{sech} x \, dx = \arctan(\sinh x) + C.^[38] For \operatorname{csch} x, the indefinite integral over $0 < x < \infty is \int \operatorname{csch} x \, dx = \ln\left|\tanh\frac{x}{2}\right| + C.^[39] These antiderivatives underscore the functions' roles in solving differential equations involving exponential decay.

Inverse Hyperbolic Functions

Logarithmic Expressions

The inverse hyperbolic functions can be expressed explicitly in terms of natural logarithms, providing closed-form representations that facilitate computation and analysis in real analysis. These expressions arise from the exponential definitions of the hyperbolic functions by solving for the inverse. For instance, the inverse hyperbolic sine, denoted \operatorname{arcsinh} x or \sinh^{-1} x, satisfies x = \sinh y, where \sinh y = \frac{e^y - e^{-y}}{2}. Substituting and solving the resulting quadratic equation in e^y yields the logarithmic form.^[15] For real x, the principal value is given by

\operatorname{arcsinh} x = \ln \left( x + \sqrt{x^2 + 1} \right),

which holds for all real x and corresponds to the branch where the result is real-valued. Similarly, for the inverse hyperbolic cosine, \operatorname{arccosh} x or \cosh^{-1} x, set x = \cosh y = \frac{e^y + e^{-y}}{2} with y \geq 0 for the principal branch, leading to

\operatorname{arccosh} x = \ln \left( x + \sqrt{x^2 - 1} \right)

for x \geq 1. The inverse hyperbolic tangent, \operatorname{arctanh} x or \tanh^{-1} x, derives from x = \tanh y = \frac{e^y - e^{-y}}{e^y + e^{-y}}, which simplifies to solving for y in terms of the ratio of exponentials, giving

\operatorname{arctanh} x = \frac{1}{2} \ln \left( \frac{1 + x}{1 - x} \right)

for |x| < 1. For the inverse hyperbolic cotangent, \operatorname{arccoth} x or \coth^{-1} x, the relation \coth y = 1 / \tanh y leads to

\operatorname{arccoth} x = \frac{1}{2} \ln \left( \frac{x + 1}{x - 1} \right)

for |x| > 1.^[40] The remaining inverse functions follow analogous derivations. The inverse hyperbolic secant, \operatorname{arcsech} x or \sech^{-1} x, uses \sech y = 1 / \cosh y, resulting in

\operatorname{arcsech} x = \ln \left( \frac{1 + \sqrt{1 - x^2}}{x} \right)

for $0 < x \leq 1.^[41] Finally, for the inverse hyperbolic cosecant, \operatorname{arccsch} x or \csch^{-1} x, from \csch y = 1 / \sinh y,

\operatorname{arccsch} x = \ln \left( \frac{1 + \operatorname{sign}(x) \sqrt{1 + x^2}}{x} \right)

for x \neq 0, ensuring the principal real branch.^[42] These logarithmic forms underscore the connection between hyperbolic and exponential functions, enabling efficient evaluation in computational contexts.^[15]

Domains and Ranges

The inverse hyperbolic functions are defined on specific domains over the real and complex numbers, with principal branches selected to ensure analyticity in cut planes. For real arguments, these functions are single-valued and real-valued on their principal domains, reflecting their roles as inverses of the corresponding hyperbolic functions. The domains and ranges vary by function, determined by the monotonicity of the hyperbolic functions and the need to avoid singularities. Over the real numbers, the principal domain and range for each inverse hyperbolic function are as follows:

Function	Domain	Range
arcsinh z	(−∞, ∞)	(−∞, ∞)
arccosh z	[1, ∞)	[0, ∞)
arctanh z	(−1, 1)	(−∞, ∞)
arccoth z	(−∞, −1) ∪ (1, ∞)	(−∞, ∞)
arcsech z	(0, 1]	[0, ∞)
arccsch z	(−∞, 0) ∪ (0, ∞)	(−∞, ∞)

These specifications ensure that the functions are bijective onto their ranges, with arcsinh, arctanh, arccoth, and arccsch being odd functions (f(−z) = −f(z)) and strictly increasing on their domains, while arccosh and arcsech are even (f(−z) = f(z)) and strictly increasing on the positive parts of their domains.^[15] All are continuous and differentiable on their interiors, with the boundaries included where the functions achieve finite limits.^[15] In the complex plane, the inverse hyperbolic functions are multivalued due to the periodicity of the exponential function underlying their logarithmic expressions, necessitating branch cuts to define principal values. The principal branch of arcsinh z has branch points at z = ±i and cuts typically along the imaginary axis segments from −i∞ to −i and from i to i∞, making it real-valued and analytic elsewhere, with Im(arcsinh z) ∈ (−π/2, π/2).^[15] For arccosh z, the principal branch features branch points at z = ±1 and a cut along (−∞, 1], ensuring it is real and nonnegative for z ≥ 1, with the range in the complex plane satisfying Re(arccosh z) ≥ 0.^[15] The arctanh z principal branch has branch points at z = ±1 and cuts along (−∞, −1] ∪ [1, ∞), analytic in the cut plane with Im(arctanh z) ∈ (−π/2, π/2) off the real axis in (−1, 1).^[15] Similarly, arccoth z shares the branch points and cuts of arctanh z (since arccoth z = arctanh(1/z)), but its principal values are chosen such that arccoth z is real and positive for z > 1, and real and negative for z < -1, on the real axis. The arcsech z has branch points at z = 0 and z = ±1, with cuts along (−∞, 0] ∪ [1, ∞), and is real and nonnegative for 0 < z ≤ 1, analogous to arccosh(1/z).^[15] For arccsch z, branch points occur at z = ±i, with cuts along the imaginary axis similar to arcsinh, and it is real-valued for all real z ≠ 0, defined as arcsinh(1/z).^[15] These principal branches are analytic in their respective cut planes and continuous up to the cuts from appropriate sides, providing a standardized framework for computations and applications in complex analysis.^[15]

Calculus Aspects

First Derivatives

The first derivatives of the hyperbolic functions are derived from their exponential definitions, yielding results analogous to those of the trigonometric functions but with distinct signs. Specifically, the derivative of the hyperbolic sine function is the hyperbolic cosine:

\frac{d}{dx} \sinh x = \cosh x

This follows from the definition \sinh x = \frac{e^x - e^{-x}}{2} and the known derivatives of the exponential functions.^[43] Similarly, the derivative of the hyperbolic cosine is the hyperbolic sine:

\frac{d}{dx} \cosh x = \sinh x

obtained via \cosh x = \frac{e^x + e^{-x}}{2}.^[43] For the hyperbolic tangent, defined as \tanh x = \frac{\sinh x}{\cosh x}, the quotient rule yields

\frac{d}{dx} \tanh x = \sech^2 x

where \sech x = \frac{1}{\cosh x}.^[44] The derivatives of the remaining functions are:

\frac{d}{dx} \coth x = -\csch^2 x, \quad \frac{d}{dx} \sech x = -\sech x \tanh x, \quad \frac{d}{dx} \csch x = -\csch x \coth x.

^[45] The derivatives of the inverse hyperbolic functions are also fundamental in calculus. For the inverse hyperbolic sine,

\frac{d}{dx} \arsinh x = \frac{1}{\sqrt{x^2 + 1}}

which holds for all real x and is derived using implicit differentiation from \sinh y = x, leading to \cosh y \cdot y' = 1 and substituting the identity \cosh^2 y - \sinh^2 y = 1.^[17] Likewise, for the inverse hyperbolic cosine (defined for x \geq 1),

\frac{d}{dx} \arcosh x = \frac{1}{\sqrt{x^2 - 1}}

obtained analogously via implicit differentiation and the hyperbolic identity.^[46] When applying the chain rule to composite functions, the derivatives retain their form scaled by the inner function's derivative. For instance, the derivative of \sinh(f(x)) is \cosh(f(x)) \cdot f'(x), mirroring the structure for \cosh(f(x)) as \sinh(f(x)) \cdot f'(x). This pattern extends naturally to other hyperbolic functions, facilitating computations in more complex expressions.^[45] Geometrically, these derivatives interpret the slopes of curves defined by hyperbolic functions, particularly in the catenary, the shape of a hanging chain under uniform gravity modeled by y = a \cosh(x/a) for some constant a > 0. The slope of this curve is \frac{dy}{dx} = \sinh(x/a), representing the tangent of the angle that the chain makes with the horizontal, which equals the ratio of arc length from the vertex to the horizontal tension component.^[47] This connection underscores the physical relevance of the derivatives in describing equilibrium shapes.^[32]

Integrals and Antiderivatives

The indefinite integrals of the primary hyperbolic functions are straightforward and mirror their differentiation rules.

\int \sinh x \, dx = \cosh x + C

\int \cosh x \, dx = \sinh x + C

These results follow from the exponential definitions of the functions or direct verification via differentiation.^[48] For the remaining hyperbolic functions, the antiderivatives involve logarithmic or inverse trigonometric expressions:

\int \tanh x \, dx = \ln |\cosh x| + C

\int \coth x \, dx = \ln |\sinh x| + C

\int \sech x \, dx = \arctan (\sinh x) + C

\int \csch x \, dx = \ln \left| \tanh \frac{x}{2} \right| + C

The integral of \tanh x arises from substitution using the identity \frac{d}{dx} \cosh x = \sinh x, while the \sech x form can be derived via the substitution u = \sinh x, leveraging \sech^2 x = 1 - \tanh^2 x. Equivalent representations for \int \sech x \, dx include $2 \arctan (\tanh (x/2)) + C. The integrals for \coth x and \csch x follow similarly from their definitions and identities.^[48]^[49] Reduction formulas facilitate evaluation of integrals involving powers of hyperbolic functions by recursively lowering the exponent. For even or odd powers greater than 1, integration by parts with the identity \cosh^2 x - \sinh^2 x = 1 (or its variants) yields:

\int \cosh^n x \, dx = \frac{\sinh x \cosh^{n-1} x}{n} + \frac{n-1}{n} \int \cosh^{n-2} x \, dx, \quad n > 1

\int \sinh^n x \, dx = -\frac{\cosh x \sinh^{n-1} x}{n} + \frac{n-1}{n} \int \sinh^{n-2} x \, dx, \quad n > 1

These formulas reduce the power by 2 each step until reaching a base case solvable by basic integrals. For odd powers, direct substitution (saving one factor for the differential) often simplifies computation without full recursion.^[50] A notable definite integral involving the hyperbolic secant is the improper integral over the real line:

\int_{-\infty}^{\infty} \sech x \, dx = \pi

This result can be established using contour integration in the complex plane, where the poles of \sech z lie on the imaginary axis, and the residue theorem applied to a suitable rectangular contour yields the value \pi.^[37]

Second Derivatives and Higher

The second derivatives of the primary hyperbolic functions follow directly from their first derivatives. For the hyperbolic sine,

\frac{d^2}{dx^2} \sinh x = \sinh x,

since the first derivative is \cosh x and the derivative of \cosh x is \sinh x.^[51] Similarly, for the hyperbolic cosine,

\frac{d^2}{dx^2} \cosh x = \cosh x,

as its first derivative is \sinh x and the second is then \cosh x.^[51] These relations highlight that both functions are eigenfunctions of the second derivative operator, satisfying y'' = y up to the sign convention inherent in their definitions.^[51] Higher-order derivatives exhibit a periodic pattern with period 4, arising from the repeated application of the differentiation rules or the characteristic differential equation y'' - y = 0. For \sinh x, the nth derivative is \sinh x when n is even and \cosh x when n is odd. For \cosh x, it is \cosh x for even n and \sinh x for odd n. This recurrence can be expressed in the complex plane as \sinh^{(n)}(x) = \sinh\left(x + n \frac{\pi i}{2}\right) (up to a phase factor), linking hyperbolic functions to rotations in the complex argument analogous to trigonometric derivatives.^[51] For inverse hyperbolic functions, higher derivatives are more involved but follow from differentiating the known first derivatives. Consider \operatorname{arcsinh} x, whose first derivative is (1 + x^2)^{-1/2}.^[52] The second derivative is then

\frac{d^2}{dx^2} \operatorname{arcsinh} x = \frac{d}{dx} \left[ (1 + x^2)^{-1/2} \right] = -\frac{1}{2} (1 + x^2)^{-3/2} \cdot 2x = -\frac{x}{(1 + x^2)^{3/2}}.

Similar expressions hold for other inverses, often involving rational functions of the form P(x) (1 \pm x^2)^{-k} where P(x) is a polynomial.^[52] These derivative properties are essential in solving linear ordinary differential equations (ODEs). The equation y'' - y = 0 has the general solution y(x) = A \cosh x + B \sinh x, where A and B are constants determined by initial conditions.^[51] Higher-order linear ODEs with constant coefficients can likewise leverage the recurrence patterns, reducing solutions to linear combinations of hyperbolic functions or their shifts. For instance, the fourth-order equation y^{(4)} - y = 0 factors into (D^2 - 1)^2 y = 0, yielding solutions involving [\sinh x](/page/Sinh), [\cosh x](/page/Cosh), x [\sinh x](/page/Sinh), and x [\cosh x](/page/Cosh).^[51]

Series and Other Representations

Taylor Series Expansions

The Taylor series expansions of the hyperbolic sine and cosine functions centered at x = 0 are infinite power series that converge for all real and complex values of x. The series for \sinh x is

\sinh x = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots,

derived from repeated differentiation of its definition \sinh x = \frac{e^x - e^{-x}}{2}. Similarly, the series for \cosh x is

\cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!} = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots,

obtained analogously from \cosh x = \frac{e^x + e^{-x}}{2}. These expansions mirror the Taylor series for \sin x and \cos x but lack alternating signs, reflecting the absence of oscillatory behavior in hyperbolic functions.^[53] The hyperbolic tangent function, defined as \tanh x = \frac{\sinh x}{\cosh x}, has a Taylor series expansion around x = 0 given by

\tanh x = \sum_{n=1}^{\infty} \frac{2^{2n} (2^{2n} - 1) B_{2n}}{(2n)!} x^{2n-1} = x - \frac{1}{3} x^3 + \frac{2}{15} x^5 - \frac{17}{315} x^7 + \cdots,

where B_{2n} denotes the $2n-th Bernoulli number (with B_2 = 1/6, B_4 = -1/30, etc.). This series arises from the quotient of the \sinh x and \cosh x expansions and converges within the radius |x| < \pi/2, limited by the poles of \tanh x at x = i(\pi/2 + k\pi) for integer k. The involvement of Bernoulli numbers highlights a connection to broader combinatorial and analytic structures, as these numbers frequently parameterize series for rational functions of exponentials.^[54]^[53] For large positive arguments, asymptotic expansions of hyperbolic functions can be obtained directly from their exponential representations, providing efficient approximations beyond the regime of power series. As x \to +\infty,

\sinh x \sim \frac{1}{2} e^x, \quad \cosh x \sim \frac{1}{2} e^x, \quad \tanh x \sim 1 - 2 e^{-2x}.

These leading-order terms capture the dominant exponential growth or saturation, with higher-order corrections available by including additional powers of e^{-2x} (for \tanh x) or e^{-2x} (for \sinh x and \cosh x). For x \to -\infty, the approximations adjust by replacing e^x with -e^{-x}/2 for \sinh x and e^{-x}/2 for \cosh x, while \tanh x \sim -1 + 2 e^{2x}. Such expansions are particularly useful for numerical computations and analytical estimates in regimes where full power series become inefficient.

Infinite Products and Continued Fractions

The infinite product representations of hyperbolic functions provide a multiplicative factorization that reveals their zeros in the complex plane. For the hyperbolic sine function, it is given by

\sinh z = z \prod_{n=1}^{\infty} \left(1 + \frac{z^2}{n^2 \pi^2}\right),

which holds for all complex z and encodes the simple zeros of \sinh z at z = n\pi i for each integer n \neq 0. This form was derived by Leonhard Euler in the 18th century as part of his work on infinite products for entire functions, drawing an analogy to the product expansion for the sine function and facilitating the study of function zeros through Weierstrass factorization principles. Similarly, the hyperbolic cosine admits the product

\cosh z = \prod_{n=1}^{\infty} \left(1 + \frac{4z^2}{(2n-1)^2 \pi^2}\right),

reflecting its zeros at z = (n - 1/2)\pi i for positive integers n. These representations converge uniformly on compact subsets of the complex plane avoiding the poles, and they parallel the Wallis product for \pi in their historical development, where Euler used similar techniques to connect products to integrals and special values.^[55]^[56] Continued fractions offer another non-power series expansion, particularly useful for computational approximations and asymptotic analysis. The hyperbolic tangent function has the continued fraction representation

\tanh z = \frac{z}{1 + \dfrac{z^2}{3 + \dfrac{z^2}{5 + \dfrac{z^2}{7 + \ddots}}}},

known as Lambert's continued fraction, which converges for all complex z in the right half-plane \Re(z) > 0 and can be extended analytically. This form, originally developed by Johann Heinrich Lambert in 1761 for the tangent function and adapted to its hyperbolic counterpart, arises from integral representations or differential equations satisfied by \tanh z, and its partial quotients grow linearly, ensuring rapid convergence compared to series expansions for moderate |z|. The structure highlights the poles of \tanh z at z = (n + 1/2)\pi i, analogous to the infinite products, and has been applied in numerical methods for evaluating hyperbolic functions near their singularities.^[54]

Extensions and Comparisons

Hyperbolic Functions of Complex Numbers

Hyperbolic functions extend naturally to complex arguments via their exponential definitions, preserving analyticity across the entire complex plane. The hyperbolic sine is defined as

\sinh z = \frac{e^z - e^{-z}}{2},

and the hyperbolic cosine as

\cosh z = \frac{e^z + e^{-z}}{2},

for any complex number z. These functions are entire, meaning they are holomorphic everywhere in the complex plane with no singularities. The remaining hyperbolic functions—tangent, cotangent, secant, and cosecant—are then expressed as ratios: \tanh z = \sinh z / \cosh z, \coth z = \cosh z / \sinh z, \sech z = 1 / \cosh z, and \csch z = 1 / \sinh z. These ratio functions inherit poles from the zeros of their denominators.^[9] The functions \sinh z and \cosh z exhibit periodicity in the complex plane with period $2\pi i, satisfying \sinh(z + 2\pi i) = \sinh z and \cosh(z + 2\pi i) = \cosh z. This follows directly from the periodicity of the exponential function, e^{z + 2\pi i} = e^z. In contrast, \tanh z has fundamental period \pi i. The zeros of \sinh z occur at z = n\pi i for integers n, leading to simple poles of \csch z at these points. Similarly, \cosh z has zeros at z = (n + 1/2)\pi i.^[10] For a complex argument z = x + i y with real x and y, the hyperbolic sine decomposes into real and imaginary parts as

\sinh(x + i y) = \sinh x \cos y + i \cosh x \sin y.

The hyperbolic cosine follows analogously:

\cosh(x + i y) = \cosh x \cos y + i \sinh x \sin y.

These identities arise from substituting the complex form into the exponential definitions and applying Euler's formula. They highlight the interplay between hyperbolic and trigonometric functions in the complex domain.^[9] The inverse hyperbolic functions, such as \sinh^{-1} z, \cosh^{-1} z, and \tanh^{-1} z, are multivalued in the complex plane and require branch cuts to define single-valued principal branches. For \sinh^{-1} z, the principal branch employs cuts along the imaginary axis from i to i\infty and from -i to -i\infty. The function \cosh^{-1} z uses cuts from -\infty to -1 and from 1 to \infty along the real axis. For \tanh^{-1} z, cuts run from -i\infty to -i and from i to i\infty on the imaginary axis. These branch structures ensure analytic continuation while respecting the multivalued nature stemming from the logarithmic expressions underlying the inverses.^[15]

Relation to Exponential and Trigonometric Functions

Hyperbolic functions are fundamentally defined in terms of exponential functions. The hyperbolic cosine is given by \cosh x = \frac{e^x + e^{-x}}{2}, and the hyperbolic sine by \sinh x = \frac{e^x - e^{-x}}{2}.^[17] These definitions extend to the other hyperbolic functions, such as \tanh x = \frac{\sinh x}{\cosh x}, which can all be expressed as combinations of e^x and e^{-x}.^[19] From these exponential forms, key identities emerge that link hyperbolic functions directly to the exponential function. For instance, e^x = \cosh x + \sinh x and e^{-x} = \cosh x - \sinh x.^[24] The fundamental identity \cosh^2 x - \sinh^2 x = 1 follows directly from substituting the exponential definitions and simplifying.^[25] Hyperbolic functions also exhibit a close analogy to trigonometric functions through complex arguments. Specifically, \cosh(ix) = \cos x and \sinh(ix) = i \sin x, where i is the imaginary unit, revealing that hyperbolic functions can be viewed as trigonometric functions evaluated at imaginary arguments.^[26] This connection underscores the structural similarities between the two sets of functions, though they differ in their geometric interpretations: trigonometric functions parameterize the unit circle in Euclidean geometry (\cos^2 x + \sin^2 x = 1), while hyperbolic functions parameterize the unit hyperbola in hyperbolic geometry (\cosh^2 x - \sinh^2 x = 1).^[18] The table below compares selected fundamental identities, highlighting the sign difference that reflects their distinct geometric roles:

Trigonometric Identity	Hyperbolic Identity
\cos^2 x + \sin^2 x = 1	\cosh^2 x - \sinh^2 x = 1
\tan^2 x + 1 = \sec^2 x	$1 - \tanh^2 x = \sech^2 x
\cot^2 x + 1 = \csc^2 x	\coth^2 x - 1 = \csch^2 x

These identities are derived analogously from their respective definitions and hold for real arguments.^[17]

Inequalities and Bounds

Hyperbolic functions satisfy several fundamental inequalities that arise from their definitions, convexity properties, and series expansions. For all real x, the hyperbolic cosine is bounded below by its minimum value: \cosh x \geq 1, with equality holding if and only if x = 0. This follows directly from the arithmetic mean-geometric mean inequality applied to the positive terms e^x and e^{-x}, since \cosh x = \frac{e^x + e^{-x}}{2} \geq \sqrt{e^x \cdot e^{-x}} = 1. Similarly, the hyperbolic sine satisfies |\sinh x| \geq |x| for all real x, with equality at x = [0](/page/0). For x > 0, this is equivalent to \frac{\sinh x}{x} \geq 1, which holds because the function \frac{\sinh x}{x} is increasing on (0, \infty) with limit 1 as x \to 0^+. The monotonicity follows from the derivative \frac{d}{dx} \left( \frac{\sinh x}{x} \right) = \frac{x \cosh x - \sinh x}{x^2} \geq 0, since x \cosh x - \sinh x \geq 0 for x \geq 0. The function \cosh x is strictly convex on \mathbb{R} because its second derivative is \frac{d^2}{dx^2} \cosh x = \cosh x > 0 for all real x. As a consequence of this convexity, Jensen's inequality applies: for any real x and y, \cosh\left( \frac{x+y}{2} \right) \leq \frac{\cosh x + \cosh y}{2}, with equality if and only if x = y. From the exponential representation \sinh x = \frac{e^x - e^{-x}}{2}, tight bounds for x > 0 are obtained by noting that e^{-x} > 0, yielding \frac{e^x}{2} - \frac{1}{2e^x} \leq \sinh x \leq \frac{e^x}{2}, where the lower bound is exact and the upper bound is strict. A useful lower bound from the Taylor series expansion of \cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots (with all terms nonnegative for real x) is \cosh x \geq 1 + \frac{x^2}{2}, with equality at x = 0. This truncation provides a quadratic approximation that underscores the function's growth behavior near the origin.

Applications

In Geometry and Special Relativity

Hyperbolic functions play a central role in the geometry of surfaces with constant negative Gaussian curvature, such as the pseudosphere, which serves as a model for hyperbolic geometry. The pseudosphere, generated by rotating a tractrix about its asymptote, has a metric that incorporates hyperbolic functions like sech and tanh in its parametric form, with coefficients E = tanh² u and G = sech² u in the first fundamental form. More generally, pseudospherical surfaces, including the single-sheet hyperboloid model of hyperbolic space, feature a line element ds² = ε R² (-dχ² + cosh² χ dφ²), where cosh χ arises naturally in the embedding coordinates, reflecting the intrinsic hyperbolic structure with curvature K = -1/R².^[57]^[58] In special relativity, hyperbolic functions parameterize the Lorentz boost transformations, providing a geometric interpretation akin to rotations in hyperbolic space. The rapidity φ is defined such that v = c tanh φ, where v is the relative velocity and c is the speed of light, allowing velocities to add linearly as φ_total = φ_1 + φ_2 under successive boosts. The Lorentz transformation for a boost along the x-direction then takes the form:

x' = x \cosh \phi - c t \sinh \phi, \quad c t' = c t \cosh \phi - x \sinh \phi,

where the Lorentz factor γ = cosh φ and γ β = sinh φ, with β = v/c, preserving the Minkowski metric.^[59]^[60] The spacetime interval in Minkowski space, ds² = -c² dt² + dx² + dy² + dz², admits a hyperbolic parametrization using proper time τ and rapidity, such as c t = c τ cosh φ and x = τ sinh φ (with y = z = 0 for simplicity), yielding ds² = -c² dτ², which highlights the timelike hyperbola's role in measuring invariant proper time along worldlines. This parametrization underscores the hyperbolic geometry of Minkowski space, where the light cone bounds causal structure.^[61] Hermann Minkowski introduced the four-dimensional spacetime framework in 1907–1908, with a lecture at the Göttingen Mathematical Society on November 5, 1907, and later at the Cologne meeting in September 1908, reformulating special relativity in terms of the invariant interval √(x² + y² + z² - c² t²) to unify space and time geometrically.^[62]

In Calculus and Differential Equations

Hyperbolic functions play a central role in solving linear second-order ordinary differential equations (ODEs) with constant coefficients, particularly those exhibiting exponential growth or decay. Consider the homogeneous equation y'' - k^2 y = 0, where k > 0 is a constant. The characteristic equation is r^2 - k^2 = 0, yielding roots r = \pm k. The general solution is then expressed as y(x) = c_1 \cosh(kx) + c_2 \sinh(kx), where c_1 and c_2 are arbitrary constants determined by initial or boundary conditions.^[21] This form leverages the definitions \cosh(kx) = \frac{e^{kx} + e^{-kx}}{2} and \sinh(kx) = \frac{e^{kx} - e^{-kx}}{2}, providing a symmetric and antisymmetric basis analogous to trigonometric functions for oscillatory equations.^[21] In boundary value problems, this solution facilitates matching conditions at endpoints. For instance, with boundary conditions y(0) = A and y(L) = B, the constants are solved as c_1 = A and c_2 = \frac{B - A \cosh(kL)}{\sinh(kL)}, ensuring the hyperbolic terms satisfy the non-oscillatory behavior inherent to the equation.^[21] Such representations are preferred over exponential forms for their compactness and utility in further manipulations, like series expansions or integral transforms.^[21] Hyperbolic functions also appear in solutions to certain nonlinear ODEs, particularly through approximation methods or exact traveling wave forms. In soliton theory, the hyperbolic tangent function \tanh often parameterizes kink or dark soliton profiles in equations like the modified Korteweg-de Vries (mKdV) equation. For the mKdV equation u_t + 6u^2 u_x + u_{xxx} = 0, a soliton solution can be u(x,t) = \frac{\sqrt{c}}{2} \tanh\left( \frac{\sqrt{c}}{2} (x - ct) \right), where c > 0 is the wave speed; this form arises via the tanh method, substituting u(\xi) = f(Y) with Y = \tanh(\xi) and \xi = k(x - ct), reducing the PDE to an ODE solvable by balancing powers.^[63] Similarly, for higher-order variants like the KdV6 equation, the tanh-coth method yields soliton solutions involving \tanh and \coth, capturing localized wave structures.^[63] For pendulum equations, nonlinear approximations sometimes invoke hyperbolic identities, though exact solutions typically require elliptic functions; however, in limiting cases or series expansions, \tanh approximations model energy transfer or separatrix behavior near unstable equilibria.^[64] The Laplace transform provides a powerful tool for incorporating hyperbolic functions into ODE solutions, especially for initial value problems. The transforms are defined as \mathcal{L}\{\sinh(at)\}(s) = \frac{a}{s^2 - a^2} and \mathcal{L}\{\cosh(at)\}(s) = \frac{s}{s^2 - a^2}, valid for \operatorname{[Re](/page/Re)}(s) > |a|.^[65] These follow directly from the exponential definitions: \sinh(at) = \frac{e^{at} - e^{-at}}{2} and \cosh(at) = \frac{e^{at} + e^{-at}}{2}, with \mathcal{L}\{e^{bt}\}(s) = \frac{1}{s - b} for \operatorname{[Re](/page/Re)}(s) > b.^[65] Applying the transform to an ODE like y'' - a^2 y = f(t) with initial conditions y(0) = y_0, y'(0) = y_1 yields an algebraic equation in the s-domain, whose solution involves these hyperbolic transforms when inverted.^[66] In partial differential equations (PDEs), hyperbolic functions integrate with Fourier series to solve boundary value problems via separation of variables. For the heat equation u_t = \alpha u_{xx} on $0 < x < L with Dirichlet boundaries u(0,t) = u(L,t) = 0 and initial u(x,0) = f(x), the solution is u(x,t) = \sum_{n=1}^\infty b_n e^{-\alpha (n\pi/L)^2 t} \sin(n\pi x / L), where b_n are Fourier sine coefficients; however, for non-homogeneous boundaries like u(0,t) = 0, u(L,t) = g(t), the steady-state or transient terms often include hyperbolic factors, such as \sinh(k(L - x)) in the eigenfunction expansion to satisfy the boundary at x = L.^[67] Similarly, in Laplace's equation \nabla^2 u = 0 on a rectangle with mixed boundaries, separated solutions take the form X(x) Y(y) = \sinh(ky) \sin(kx), with coefficients from Fourier series of the boundary data, ensuring orthogonality and convergence.^[67] This combination exploits the hyperbolic growth to match asymmetric boundaries while the trigonometric part handles periodicity.^[68]

In Engineering and Physics

Hyperbolic functions play a crucial role in electrical engineering, particularly in the analysis of transmission lines, where they describe wave propagation and voltage/current distributions. The general solution for the voltage V(x) at a distance x from the receiving end of a uniform transmission line is expressed as

V(x) = V_R \cosh(\gamma x) + Z_c I_R \sinh(\gamma x),

where V_R and I_R are the voltage and current at the receiving end, Z_c is the characteristic impedance, and \gamma is the propagation constant. This form arises from solving the second-order telegrapher's equations, which model the distributed inductance, capacitance, resistance, and conductance along the line.^[69] Similar expressions apply to the current, enabling engineers to compute input impedances and reflection coefficients essential for designing high-frequency circuits and power systems. In wave mechanics and optics, hyperbolic functions appear in soliton solutions to nonlinear partial differential equations that govern phenomena like optical pulse propagation in fibers. For the Korteweg-de Vries (KdV) equation, which models shallow-water waves and certain plasma waves, the one-soliton solution takes the form

u(x, t) = -\frac{c}{2} \sech^2 \left( \sqrt{\frac{c}{2}} (x - c t) \right),

where c > 0 is the wave speed; this profile maintains its shape during propagation due to a balance between nonlinearity and dispersion.^[70] Likewise, in the focusing nonlinear Schrödinger equation, relevant to modulated wave envelopes in nonlinear media, the fundamental bright soliton is

\psi(x, t) = \eta \sech \left( \eta (x - v t) \right) \exp \left( i \left( k x - \omega t \right) \right),

with amplitude \eta, velocity v = 2k, and frequency \omega = k^2 - \eta^2 / 2; this solution describes stable, non-dispersive pulses in optical solitons.^[71] In thermodynamics, hyperbolic functions facilitate computations in statistical mechanics, particularly for fermionic systems. In certain quantum field theory models, such as those for Chern-Simons-matter theories, matrix representations of partition functions incorporate hyperbolic cosine terms.^[72] In signal processing, the hyperbolic tangent function serves as a common activation in neural networks, providing smooth, bounded outputs that aid gradient-based training while mitigating vanishing gradient issues compared to sigmoidal alternatives.^[73] Its derivative, \sech^2, naturally arises in backpropagation.