Hyperbolic functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t), the derivatives of sinh(t) and cosh(t) are cosh(t) and sinh(t).

Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations in special relativity. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, and fluid dynamics.

The basic hyperbolic functions are:

• hyperbolic sine "sinh" (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/),
• hyperbolic cosine "cosh" (/ˈkɒʃ, ˈkoʊʃ/),

from which are derived:

• hyperbolic tangent "tanh" (/ˈtæŋ, ˈtæntʃ, ˈθæn/),
• hyperbolic cotangent "coth" (/ˈkɒθ, ˈkoʊθ/),
• hyperbolic secant "sech" (/ˈsɛtʃ, ˈʃɛk/),
• hyperbolic cosecant "csch" or "cosech" (/ˈkoʊsɛtʃ, ˈkoʊʃɛk/)

corresponding to the derived trigonometric functions.

The inverse hyperbolic functions are:

• inverse hyperbolic sine "arsinh" (also denoted "sinh", "asinh" or sometimes "arcsinh")
• inverse hyperbolic cosine "arcosh" (also denoted "cosh", "acosh" or sometimes "arccosh")
• inverse hyperbolic tangent "artanh" (also denoted "tanh", "atanh" or sometimes "arctanh")
• inverse hyperbolic cotangent "arcoth" (also denoted "coth", "acoth" or sometimes "arccoth")
• inverse hyperbolic secant "arsech" (also denoted "sech", "asech" or sometimes "arcsech")
• inverse hyperbolic cosecant "arcsch" (also denoted "arcosech", "csch", "cosech","acsch", "acosech", or sometimes "arccsch" or "arccosech")

Condensed: the full section is in Wikipedia.

The first known calculation of a hyperbolic trigonometry problem is attributed to Gerardus Mercator when issuing the Mercator map projection circa 1566. It requires tabulating solutions to a transcendental equation involving hyperbolic functions.

The first to suggest a similarity between the sector of the circle and that of the hyperbola was Isaac Newton in his 1687 Principia Mathematica.

Roger Cotes suggested to modify the trigonometric functions using the imaginary unit \(i=\sqrt{-1}\) to obtain an oblate spheroid from a prolate one.

Hyperbolic functions were formally introduced in 1757 by Vincenzo Riccati. Riccati used Sc. and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. (sinus/cosinus hyperbolico) to refer to hyperbolic functions. As early as 1759, Daviet de Foncenex showed the interchangeability of the trigonometric and hyperbolic functions using the imaginary unit and extended de Moivre's formula to hyperbolic functions.

During the 1760s, Johann Heinrich Lambert systematized the use functions and provided exponential expressions in various publications. Lambert credited Riccati for the terminology and names of the functions, but altered the abbreviations to those used today.

With hyperbolic angle u, the hyperbolic functions sinh and cosh can be defined with the exponential function e. In the figure \(A =(e^{-u}, e^u), \ B=(e^u, \ e^{-u}), \ OA + OB = OC\) .

• Hyperbolic sine: the odd part of the exponential function, that is, \[\sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x}.\]
• Hyperbolic cosine: the even part of the exponential function, that is, \[\cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x}.\]

• Hyperbolic tangent: \[\tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1}.\]
• Hyperbolic cotangent: for x ≠ 0, \[\coth x = \frac{\cosh x}{\sinh x} = \frac {e^x + e^{-x}} {e^x - e^{-x}} = \frac{e^{2x} + 1} {e^{2x} - 1}.\]
• Hyperbolic secant: \[\operatorname{sech} x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}} = \frac{2e^x} {e^{2x} + 1}.\]
• Hyperbolic cosecant: for x ≠ 0, \[\operatorname{csch} x = \frac{1}{\sinh x} = \frac {2} {e^x - e^{-x}} = \frac{2e^x} {e^{2x} - 1}.\]

The hyperbolic functions may be defined as solutions of differential equations: The hyperbolic sine and cosine are the solution (s, c) of the system \[\begin{aligned} c'(x)&=s(x),\\ s'(x)&=c(x),\\ \end{aligned}\] with the initial conditions \(s(0) = 0, c(0) = 1.\) The initial conditions make the solution unique; without them any pair of functions \((a e^x + b e^{-x}, a e^x - b e^{-x})\) would be a solution.

sinh(x) and cosh(x) are also the unique solution of the equation f ″(x) = f (x), such that f (0) = 1, f ′(0) = 0 for the hyperbolic cosine, and f (0) = 0, f ′(0) = 1 for the hyperbolic sine.

Hyperbolic functions may also be deduced from trigonometric functions with complex arguments:

• Hyperbolic sine: \[\sinh x = -i \sin (i x).\]
• Hyperbolic cosine: \[\cosh x = \cos (i x).\]
• Hyperbolic tangent: \[\tanh x = -i \tan (i x).\]
• Hyperbolic cotangent: \[\coth x = i \cot (i x).\]
• Hyperbolic secant: \[\operatorname{sech} x = \sec (i x).\]
• Hyperbolic cosecant:\[\operatorname{csch} x = i \csc (i x).\]

where i is the imaginary unit with i = −1.

The above definitions are related to the exponential definitions via Euler's formula (See § Hyperbolic functions for complex numbers below).

It can be shown that the area under the curve of the hyperbolic cosine (over a finite interval) is always equal to the arc length corresponding to that interval: \[\text{area} = \int_a^b \cosh x \,dx = \int_a^b \sqrt{1 + \left(\frac{d}{dx} \cosh x \right)^2} \,dx = \text{arc length.}\]

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule (named after George Osborn) states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for \(\theta\), \(2\theta\), \(3\theta\) or \(\theta\) and \(\varphi\) into a hyperbolic identity, by:

1. expanding it completely in terms of integral powers of sines and cosines,
2. changing sine to sinh and cosine to cosh, and
3. switching the sign of every term containing a product of two sinhs.

Odd and even functions: \[\begin{aligned} \sinh (-x) &= -\sinh x \\ \cosh (-x) &= \cosh x \\ \tanh (-x) &= -\tanh x \\ \coth (-x) &= -\coth x \\ \operatorname{sech} (-x) &= \operatorname{sech} x \\ \operatorname{csch} (-x) &= -\operatorname{csch} x \end{aligned}\]

Reciprocals:

\[\begin{aligned} \operatorname{arsech} x &= \operatorname{arcosh} \left(\frac{1}{x}\right) \\ \operatorname{arcsch} x &= \operatorname{arsinh} \left(\frac{1}{x}\right) \\ \operatorname{arcoth} x &= \operatorname{artanh} \left(\frac{1}{x}\right) \end{aligned}\]

Analogous to Euler's formula:

\[\begin{aligned} \cosh x + \sinh x &= e^x \\ \cosh x - \sinh x &= e^{-x} \end{aligned}\]

Analogous to the Pythagorean trigonometric identity:

Condensed: the full section is in Wikipedia.

\[\begin{aligned} \sinh(x + y) &= \sinh x \cosh y + \cosh x \sinh y \\ \cosh(x + y) &= \cosh x \cosh y + \sinh x \sinh y \\ \tanh(x + y) &= \frac{\tanh x +\tanh y}{1+ \tanh x \tanh y } \\ \sinh(x - y) &= \sinh x \cosh y - \cosh x \sinh y \\ \cosh(x - y) &= \cosh x \cosh y - \sinh x \sinh y \\ \tanh(x - y) &= \frac{\tanh x -\tanh y}{1- \tanh x \tanh y } \\ \end{aligned}\] particularly \[\begin{aligned} \cosh (2x) &= \sinh^2{x} + \cosh^2{x} = 2\sinh^2 x + 1 = 2\cosh^2 x - 1 \\ \sinh (2x) &= 2\sinh x \cosh x \\ \tanh (2x) &= \frac{2\tanh x}{1+ \tanh^2 x } \\ \end{aligned}\]

\[\begin{aligned} \sinh x + \sinh y &= 2 \sinh \left(\frac{x+y}{2}\right) \cosh \left(\frac{x-y}{2}\right)\\ \cosh x + \cosh y &= 2 \cosh \left(\frac{x+y}{2}\right) \cosh \left(\frac{x-y}{2}\right)\\ \sinh x - \sinh y &= 2 \cosh \left(\frac{x+y}{2}\right) \sinh \left(\frac{x-y}{2}\right)\\ \cosh x - \cosh y &= 2 \sinh \left(\frac{x+y}{2}\right) \sinh \left(\frac{x-y}{2}\right)\\ \end{aligned}\]

\[\begin{aligned} \cosh x\, \cosh y &= \tfrac12\bigl(\!\!~\cosh(x + y) + \cosh(x - y)\bigr) \\[5mu] \sinh x\, \sinh y &= \tfrac12\bigl(\!\!~\cosh(x + y) - \cosh(x - y)\bigr) \\[5mu] \sinh x\, \cosh y &= \tfrac12\bigl(\!\!~\sinh(x + y) + \sinh(x - y)\bigr) \\[5mu] \cosh x\, \sinh y &= \tfrac12\bigl(\!\!~\sinh(x + y) - \sinh(x - y)\bigr) \\[5mu] \end{aligned}\]

\[\begin{aligned} \sinh\left(\frac{x}{2}\right) &= \frac{\sinh x}{\sqrt{2 (\cosh x + 1)} } &&= \sgn x \, \sqrt \frac{\cosh x - 1}{2} \\[6px] \cosh\left(\frac{x}{2}\right) &= \sqrt \frac{\cosh x + 1}{2}\\[6px] \tanh\left(\frac{x}{2}\right) &= \frac{\sinh x}{\cosh x + 1} &&= \sgn x \, \sqrt \frac{\cosh x-1}{\cosh x+1} = \frac{e^x - 1}{e^x + 1} \end{aligned}\]

where sgn is the sign function.

If x ≠ 0 then

\[\tanh\left(\frac{x}{2}\right) = \frac{\cosh x - 1}{\sinh x} = \coth x - \operatorname{csch} x\]

When ⁠\(t = \tanh\left(\frac{x}{2}\right)\)⁠, \[\begin{aligned} & \sinh x = \frac{2t}{1 - t^2}, & & \cosh x = \frac{1 + t^2}{1 - t^2}, \\[8pt] & \tanh x = \frac{2t}{1 + t^2}, & & \coth x = \frac{1 + t^2}{2t}, \\[8pt] & \operatorname{sech} x = \frac{1 - t^2}{1 + t^2}, & & \operatorname{csch} x = \frac{1 - t^2}{2t}, \\[8pt] & & \quad dx = \frac{2}{t^2-1}\, dt. \end{aligned}\]

The following inequality is useful in statistics: \[\operatorname{cosh}(t) \leq e^{t^2 /2}.\]

It can be proved by comparing the Taylor series of the two functions term by term.