Sinh
The hyperbolic sine function, denoted \sinh(x), is a mathematical function defined as \sinh(x) = \frac{e^{x} - e^{-x}}{2}, where e is the base of the natural logarithm.[1] This expression links it directly to exponential functions, distinguishing it from trigonometric functions while sharing analogous properties.[2] As an odd function, \sinh(-x) = -\sinh(x), and it satisfies \sinh(0) = 0, with its graph asymptotic to e^{x}/2 for positive x and -e^{-x}/2 for negative x.[1] Introduced in the mid-18th century, hyperbolic functions like sinh were developed independently by Italian mathematician Vincenzo Riccati around 1757–1762, who connected them to the unit hyperbola, and by Swiss polymath Johann Heinrich Lambert in the 1760s.[3] Lambert formalized the notation "sinh" by 1771, abbreviating "sinus hyperbolicus" to reflect its role as the hyperbolic counterpart to the sine function.[4] These functions satisfy key identities, such as \cosh^{2}(x) - \sinh^{2}(x) = 1, mirroring the Pythagorean theorem but for hyperbolas, and \sinh(2x) = 2 \sinh(x) \cosh(x).[1] In applications, sinh appears in physics and engineering, notably in describing the shape of suspended cables, chains, or electrical transmission lines under uniform gravity, known as the catenary curve, where the hyperbolic cosine (cosh) is primary but sinh features in parametrizations and derivatives.[5] It also models the dynamics of certain mechanical systems.[2]Definition and Notation
Explicit Formula
The hyperbolic sine function, commonly denoted as \sinh z, is explicitly defined for any complex number z by the formula \sinh z = \frac{e^z - e^{-z}}{2}. [6][7] This expression originates from the exponential function, where \sinh z captures the antisymmetric (odd) component of e^z relative to the origin, specifically as half the difference between e^z and its reciprocal e^{-z}, ensuring \sinh 0 = 0 and facilitating analogies with trigonometric functions.[6][8] For real arguments x \in \mathbb{R}, the formula simplifies to the identical form \sinh x = \frac{e^x - e^{-x}}{2}, which serves as the primary computational basis for the function in real analysis.[6][7] In standard mathematical notation, particularly for real inputs, the function is often written without parentheses as \sinh x, a convention that aligns with practices for elementary functions like \sin x and \cos x.[7]Relation to Exponential Function
The hyperbolic sine function, denoted \sinh x, is fundamentally related to the exponential function through the definition \sinh x = \frac{e^x - e^{-x}}{2}.[2] This expression arises as the odd component of the exponential function, separating the antisymmetric contributions from e^x and e^{-x}.[9] This definition was first introduced in 1757 by Italian mathematician and Jesuit priest Vincenzo Riccati in his work Opuscula ad res physicas et mathematicas pertinentes, where he denoted it as "Sh x" and presented it as an analog to the sine function but based on exponential rather than circular arguments.[3] Riccati developed the addition formulas for these functions, establishing their role in solving differential equations and geometric problems involving hyperbolas.[10] The connection to exponentials is rigorously demonstrated through Taylor series expansions. The series for e^x is \sum_{n=0}^{\infty} \frac{x^n}{n!}, and for e^{-x} it is \sum_{n=0}^{\infty} \frac{(-x)^n}{n!} = \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n!}. Subtracting these yields: e^x - e^{-x} = \sum_{n=0}^{\infty} \frac{x^n - (-1)^n x^n}{n!}. For even n, the terms cancel since (-1)^n = 1; for odd n, (-1)^n = -1, so x^n - (-1)^n x^n = 2x^n, resulting in: e^x - e^{-x} = 2 \sum_{k=0}^{\infty} \frac{x^{2k+1}}{(2k+1)!}. Dividing by 2 gives the Taylor series for \sinh x: \sinh x = \sum_{k=0}^{\infty} \frac{x^{2k+1}}{(2k+1)!} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots, which converges for all real x and confirms the exponential origin without relying on differential equations.[11][12] For numerical computation, the exponential definition offers advantages when |x| is large. When x \gg 1, e^{-x} becomes negligible compared to e^x, so \sinh x \approx \frac{e^x}{2}; similarly, for x \ll -1, \sinh x \approx -\frac{e^{-x}}{2} = \frac{\operatorname{sign}(x) e^{|x|}}{2}. This approximation exploits the dominance of one exponential term, reducing potential overflow issues in direct evaluation by focusing on the larger exponent while treating the smaller as zero.[1] Such methods are essential in scientific computing libraries for handling extreme values without loss of precision.[13]Fundamental Properties
Domain, Range, and Periodicity
The hyperbolic sine function, denoted \sinh z, is defined for all complex numbers z \in \mathbb{C}, making its domain the entire complex plane.[6] As an entire function, \sinh z is analytic everywhere in \mathbb{C}.[6] For real inputs x \in \mathbb{R}, the domain of \sinh x is all real numbers (-\infty, +\infty), and its range is also all real numbers (-\infty, +\infty), establishing a bijective mapping from \mathbb{R} to \mathbb{R}.[14] This bijectivity follows from its relation to the exponential function, where \sinh x = \frac{e^x - e^{-x}}{2} ensures a one-to-one correspondence.[14] On the real line, \sinh x is strictly increasing, with \sinh 0 = 0 serving as a fixed point.[14] Unlike the trigonometric sine function, which is periodic over the reals, \sinh z has no real period.[6] However, it exhibits periodicity in the complex plane with period $2\pi i, satisfying \sinh([z](/page/Z) + 2\pi i) = \sinh [z](/page/Z) for all z \in \mathbb{C}.[6] For complex inputs, the range of \sinh [z](/page/Z) is the entire complex plane \mathbb{C}, as the function is surjective.[6]Symmetry and Even-Odd Behavior
The hyperbolic sine function, defined over the real numbers, exhibits odd parity, satisfying \sinh(-x) = -\sinh(x) for all real x. This property follows directly from the exponential definition \sinh x = \frac{e^x - e^{-x}}{2}; substituting -x yields \sinh(-x) = \frac{e^{-x} - e^x}{2} = -\frac{e^x - e^{-x}}{2} = -\sinh x.[15] As a consequence of its odd nature, the graph of \sinh x demonstrates point symmetry, or reflection symmetry across the origin (0,0) in the real plane, meaning that the point (x, \sinh x) maps to (-x, -\sinh x).[15] Regarding asymptotic behavior, \sinh x \to \infty as x \to \infty and \sinh x \to -\infty as x \to -\infty, reflecting the dominant exponential growth in the positive direction and decay in the negative, approximated by \sinh x \approx \frac{e^x}{2} for large positive x.[16][15] In the complex domain, \sinh z preserves conjugation symmetry, such that \sinh(\bar{z}) = \overline{\sinh z} for any complex z, where \bar{z} denotes the complex conjugate; this holds because the defining exponentials satisfy e^{\bar{z}} = \overline{e^z}, ensuring the overall expression conjugates accordingly. For z = x + iy with real x, y, this manifests in the expansion \sinh z = \sinh x \cos y + i \cosh x \sin y, whose conjugate matches \sinh(x - iy).[15]Calculus and Analysis
Derivatives
The first derivative of the hyperbolic sine function is given by \frac{d}{dx} \sinh x = \cosh x. This follows from the exponential definition \sinh x = \frac{e^x - e^{-x}}{2}, where differentiation yields \frac{d}{dx} \sinh x = \frac{e^x + e^{-x}}{2} = \cosh x.[7][17] Higher-order derivatives of \sinh x exhibit a periodic pattern due to the repeated application of the basic differentiation rules. Specifically, the nth derivative is \frac{d^n}{dx^n} \sinh x = \sinh x when n is even, and \frac{d^n}{dx^n} \sinh x = \cosh x when n is odd and greater than zero. This alternation arises from the exponential form: \frac{d^n}{dx^n} \sinh x = \frac{1}{2} \left( e^x - (-1)^n e^{-x} \right), which simplifies to \sinh x for even n (since (-1)^n = 1) and \cosh x for odd n (since (-1)^n = -1). The companion function \cosh x thus plays a key role in this sequence.[7] The Taylor series expansion of \sinh x around x = 0 is \sinh x = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots, which directly encodes the higher derivatives through the general Taylor formula \sinh x = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!} x^k, where f^{(k)}(0) alternates between 0 (even k > 0) and 1 (odd k). This series confirms the derivative pattern, as the coefficients reflect evaluations of \sinh x and \cosh x at the origin.[7][18] In the complex domain, \sinh z is an entire function, analytic everywhere, with its derivative \frac{d}{dz} \sinh z = \cosh z holding by the same exponential definition and satisfying the Cauchy-Riemann equations throughout the complex plane; thus, analytic continuation preserves the real differentiation rules without modification.[7][19]Integrals and Antiderivatives
The indefinite integral of the hyperbolic sine function is given by \int \sinh x \, dx = \cosh x + C, where C is the constant of integration. This result follows directly from the definition of the hyperbolic functions in terms of exponentials, \sinh x = \frac{e^x - e^{-x}}{2} and [\cosh x](/page/Cosh) = \frac{e^x + e^{-x}}{2}, by integrating term by term. Verification is obtained by differentiation: the derivative of [\cosh x](/page/Cosh) is \sinh x, confirming the antiderivative.[20] Hyperbolic substitutions provide an effective method for evaluating integrals involving square roots of quadratic expressions, particularly those resembling the form \sqrt{x^2 + a^2}. For instance, the substitution x = a \sinh u (with dx = a \cosh u \, du) transforms the integral \int \frac{dx}{\sqrt{x^2 + a^2}} into \int \frac{a \cosh u \, du}{a \cosh u} = \int du = u + C = \sinh^{-1} \left( \frac{x}{a} \right) + C, since \cosh^2 u - \sinh^2 u = 1 ensures \sqrt{x^2 + a^2} = a \cosh u. This approach simplifies the integration and yields the inverse hyperbolic sine function as the antiderivative, which is equivalent to \ln \left( x + \sqrt{x^2 + a^2} \right) + C via logarithmic identities. Such substitutions are particularly useful in applications like finding arc lengths or areas under hyperbolas.[20] Definite integrals involving \sinh x often arise in transforms and series expansions, providing closed-form evaluations in specific contexts. A representative example is the Laplace transform of \sinh x, \int_0^\infty e^{-s x} \sinh x \, dx = \frac{1}{s^2 - 1}, \quad \Re(s) > 1, which is derived by expressing \sinh x in exponential form and integrating term by term, yielding a rational function useful for solving initial-value problems in differential equations. This integral establishes the scale of convergence and highlights the role of \sinh x in transform methods for unbounded domains.[21] For approximations, the Taylor series expansion of \sinh x around zero, \sinh x = \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}, allows term-by-term integration to obtain \int \sinh x \, dx = \sum_{n=0}^\infty \frac{x^{2n+2}}{(2n+2) (2n+1)!} + C. This series converges for all real x and provides a polynomial approximation for numerical evaluation, especially near x = 0, where the first few terms suffice for high accuracy in computational contexts.[7] The antiderivatives involving \sinh x play a key role in solving linear homogeneous differential equations with constant coefficients, such as y'' - y = 0, whose general solution is y = A \cosh x + B \sinh x. Integrating both sides or using variation of parameters in related nonhomogeneous equations often leads to expressions with \cosh x as the antiderivative of \sinh x, facilitating solutions in physical models like vibrating strings or hanging chains (catenaries).Identities and Relations
Hyperbolic Pythagorean Identities
The fundamental hyperbolic Pythagorean identity is given by \cosh^2 x - \sinh^2 x = 1, where \cosh x = \frac{e^x + e^{-x}}{2} and \sinh x = \frac{e^x - e^{-x}}{2}.[1] This identity is derived by substituting the exponential definitions into the left-hand side: \cosh^2 x = \left( \frac{e^x + e^{-x}}{2} \right)^2 = \frac{e^{2x} + 2 + e^{-2x}}{4}, \sinh^2 x = \left( \frac{e^x - e^{-x}}{2} \right)^2 = \frac{e^{2x} - 2 + e^{-2x}}{4}, so \cosh^2 x - \sinh^2 x = \frac{(e^{2x} + 2 + e^{-2x}) - (e^{2x} - 2 + e^{-2x})}{4} = \frac{4}{4} = 1. [1] This algebraic verification confirms the identity for all real x.[22] From the fundamental identity, several derived forms follow directly. For instance, rearranging yields \sinh^2 x = \cosh^2 x - 1 and $1 + \sinh^2 x = \cosh^2 x. These variants are obtained by simple algebraic manipulation of the core equation.[23] The identity generalizes to complex arguments, holding for all complex numbers z \in \mathbb{C}, as the exponential definitions extend analytically to the entire complex plane and the algebraic proof remains valid.[24]Connections to Trigonometric Functions
The hyperbolic sine function exhibits direct analogies to the trigonometric sine and cosine functions through transformations involving imaginary arguments in the complex plane. A fundamental relation is given by \sinh(iz) = i \sin z, where i is the imaginary unit, establishing that the hyperbolic sine of an imaginary argument yields the trigonometric sine scaled by i. Conversely, \sinh z = -i \sin(iz). These identities arise from the exponential definitions of both functions and underscore the unified treatment of hyperbolic and trigonometric functions within complex analysis. A parallel connection exists for the hyperbolic cosine: \cosh z = \cos(iz). This mapping links the hyperbolic functions, associated with hyperbolas, to the circular functions tied to circles, via rotation by 90 degrees in the complex plane (multiplication by i). Such relations allow hyperbolic identities to be derived from trigonometric ones by substituting imaginary arguments, and vice versa, facilitating proofs and computations across both systems. These connections are further illuminated by extensions of Euler's formula. For trigonometric functions, e^{ix} = \cos x + i \sin x, capturing periodic oscillation, while the hyperbolic analog is e^{x} = \cosh x + \sinh x, reflecting exponential growth and decay. The imaginary exponent in the trigonometric case contrasts with the real exponent in the hyperbolic case, yet both stem from the exponential function, providing a common foundation that explains the structural similarities between the two function families. In applications, particularly within complex analysis, these relations enable the solution of trigonometric equations using hyperbolic functions. For example, solving \sin z = k for complex z transforms via the identity to \sinh(iz) = i k, yielding z = -i \arcsinh(i k) + 2\pi n or z = \pi + i \arcsinh(i k) + 2\pi n for integers n, which is especially useful when |k| > 1 since the real sine is bounded but extends unboundedly in the complex domain. This approach leverages the real-valued nature of hyperbolic functions for numerical and analytical insights into complex trigonometric solutions.[25]Graphical Representation and Values
Graph Characteristics
The graph of the hyperbolic sine function, \sinh x, exhibits an odd symmetry, forming a smooth S-shaped curve that passes through the origin./07%3A_Analytic_Geometry_and_Plane_Curves/7.05%3A_Hyperbolic_Functions) This curve is strictly increasing for all real x, with \sinh 0 = 0 marking the origin as a point of inflection where the concavity changes.[16] For x > 0, the function is concave up, reflecting its exponential growth, while the second derivative \sinh x > 0 confirms this behavior./07%3A_Analytic_Geometry_and_Plane_Curves/7.05%3A_Hyperbolic_Functions) Asymptotically, \sinh x approaches \frac{e^x}{2} as x \to +\infty, dominating the negative exponential term, and \sinh x \approx -\frac{e^{-x}}{2} as x \to -\infty, due to the symmetry and the form \sinh x = \frac{e^x - e^{-x}}{2}.[1] These asymptotes highlight the unbounded nature of the function, extending to +\infty and -\infty without horizontal bounds, unlike the bounded S-shape of \tanh x or the pole-disrupted graph of \coth x.[2] In the complex plane, \sinh z is an entire function, holomorphic everywhere with no poles or branch cuts, allowing its graph to extend analytically across the Argand plane.[26] Level curves, or hyperbolic contours, of \operatorname{Re}(\sinh z) and \operatorname{Im}(\sinh z) form confocal hyperbolas, illustrating the conformal mapping properties that distort the plane into hyperbolic geometries.[27]Numerical Values and Tables
The hyperbolic sine function, sinh(x), yields exact value 0 at x = 0, and for positive x, it increases monotonically from there. Computed values at integer points up to 5 are provided in the following table, rounded to six decimal places for practicality; these can be obtained via the definition sinh(x) = (e^x - e^{-x})/2 or high-precision libraries. Note the odd symmetry: sinh(-x) = -sinh(x).| x | sinh(x) |
|---|---|
| 0 | 0.000000 |
| 1 | 1.175201 |
| 2 | 3.626861 |
| 3 | 10.017875 |
| 4 | 27.289917 |
| 5 | 74.203210 |
| -1 | -1.175201 |
| -2 | -3.626861 |
| -3 | -10.017875 |
| -4 | -27.289917 |
| -5 | -74.203210 |
| y | arcsinh(y) |
|---|---|
| 0 | 0.000000 |
| 1 | 0.881374 |
| 2 | 1.443635 |
| 3 | 1.818446 |
| 4 | 2.094712 |
| 5 | 2.312438 |