Hyperbolic Function

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

[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.


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1999

[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.