Hyperbolic Tangent (tanh) Function

A Hyperbolic Tangent (tanh) Function is a hyperbolic function based on a tangent function.

• Context:
  • It can be defined as [math]\displaystyle{ \tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1} = \frac{1 - e^{-2x}} {1 + e^{-2x}} }[/math].
• Example(s):
  • a Hyperbolic Tangent Activation Function.
  • …
• Counter-Example(s):
  • a tan() Function.
  • a Hyperbolic Cosine Function, ...
  • a Logistic Sigmoid Function.
• See: Inverse Hyperbolic Function, Hyperbolic Angle, Hyperbolic Sector.

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References

2017a

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/hyperbolic_function#Hyperbolic_tangent Retrieved:2017-12-3.
  • The hyperbolic tangent is the solution to the differential equation [math]\displaystyle{ f'=1-f^2 }[/math] with f(0)=0 and the nonlinear boundary value problem:  : [math]\displaystyle{ \frac{1}{2} f'' = f^3 - f ; \quad f(0) = f'(\infty) = 0 }[/math]

2015

• http://en.wikipedia.org/wiki/Hyperbolic_function#Standard_algebraic_expressions
  • [math]\displaystyle{ \tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1} = \frac{1 - e^{-2x}} {1 + e^{-2x}} }[/math]

2014

• http://mathworld.wolfram.com/HyperbolicTangent.html
  • QUOTE: As Gauss showed in 1812, the hyperbolic tangent can be written using a continued fraction as [math]\displaystyle{ tanh_x=x/(1+(x^2)/(3+(x^2)/(5+...))) (12) }[/math] (Wall 1948, p. 349; Olds 1963, p. 138). This continued fraction is also known as Lambert's continued fraction (Wall 1948, p. 349).

    The hyperbolic tangent tanhx satisfies the second-order ordinary differential equation [math]\displaystyle{ \frac{1}{2} f'' = f^3 - f }[/math] together with the boundary conditions [math]\displaystyle{ f(0)=0 }[/math] and [math]\displaystyle{ f'(\infty)=0 }[/math].

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