Hyperbolic Function

A Hyperbolic Function is based on its corresponding trigonometric function when expressed in terms of complex exponential function.

• Context:
  • It can be defined by removing the [math]\displaystyle{ i }[/math] appearing in the in the trigonometric functions expressed in complex exponential. For instance:

[math]\displaystyle{ \sin(x)= \frac{1}{2i}\left(e^{ix}-e^{-ix}\right) \Longrightarrow \sinh(x)= \frac{1}{2}\left(e^{x}-e^{-x}\right) }[/math]

where [math]\displaystyle{ \sin(x) }[/math] is the sine function, [math]\displaystyle{ \sinh(x) }[/math] is the hyperbolic sine function and [math]\displaystyle{ x }[/math] is a real number. The trigonometric functions expressed in terms of complex exponentials are derived from Euler's Formula.

• Example(s):
  • Hyperbolic Tangent Function.
  • Hyperbolic Sine Function.
  • Hyperbolic Cosine Function.
• Counter-Example(s):
  • Tangent Function.
  • Sine Function.
  • Cosine Function.
• See: Hyperbolic Sine Function, Hyperbolic Tangent Function, Complex Exponential Function, Hyperbolic Cosine Function.

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References

2017

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Hyperbolic_function Retrieved:2017-12-3.
  • In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.

    The basic hyperbolic functions are the hyperbolic sine "sinh" (or ), and the hyperbolic cosine "cosh" from which are derived the hyperbolic tangent "tanh" (or ), hyperbolic cosecant "csch" or "cosech" (or ), hyperbolic secant "sech" (or ), and hyperbolic cotangent "coth" (or ), corresponding to the derived trigonometric functions. The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also denoted "sinh⁻¹", "asinh" or sometimes "arcsinh") and so on. 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 equilateral hyperbola. The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. Hyperbolic functions occur in the solutions of many linear differential equations (for example, the equation defining a catenary), of some cubic equations, in calculations of angles and distances in hyperbolic geometry, and of Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. When considered defined by a complex variable, the hyperbolic functions are rational functions of exponentials, and are hence holomorphic.

    Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.

1999

• (Wolfram Mathworld , 1999) ⇒ http://mathworld.wolfram.com/HyperbolicFunctions.html
  • QUOTE: The hyperbolic functions [math]\displaystyle{ \sinh\;z, \cosh\;z, \tanh\;z, \textrm{csch}\;z, \textrm{sech}\;z, \coth\;z }[/math] (hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, and hyperbolic cotangent) are analogs of the circular functions, defined by removing [math]\displaystyle{ i }[/math]'s the complex exponentials. For example,

[math]\displaystyle{ \cos z=1/2(e^{iz}+e^{-iz}),\quad \textrm{so} \quad \cosh z=1/2(e^z+e^{-z}) }[/math].

(...) The hyperbolic functions share many properties with the corresponding circular functions. In fact, just as the circle can be represented parametrically by

[math]\displaystyle{ x =a\cos t \quad y = a \sin t }[/math]

a rectangular hyperbola (or, more specifically, its right branch) can be analogously represented by

[math]\displaystyle{ x =a\cosh t \quad y=a \sinh t }[/math]

where [math]\displaystyle{ \cosh t }[/math] is the hyperbolic cosine and [math]\displaystyle{ \sinh t }[/math] is the hyperbolic sine.

The hyperbolic functions arise in many problems of mathematics and mathematical physics in which integrals involving [math]\displaystyle{ \sqrt{1+x^2} }[/math] arise (whereas the circular functions involve [math]\displaystyle{ \sqrt{1-x^2} }[/math]. For instance, the hyperbolic sine arises in the gravitational potential of a cylinder and the calculation of the Roche limit. The hyperbolic cosine function is the shape of a hanging cable (the so-called catenary). The hyperbolic tangent arises in the calculation of and rapidity of special relativity. All three appear in the Schwarzschild metric using external isotropic Kruskal coordinates in general relativity. The hyperbolic secant arises in the profile of a laminar jet. The hyperbolic cotangent arises in the Langevin function for magnetic polarization.

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