Hyperbolic Sine Function

A Hyperbolic Sine Function is a hyperbolic function based on the sine function.

• AKA: Hyperbolic Sine.
• Context:
  • It is defined as

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

where [math]\displaystyle{ \sin(x) }[/math] is the sine function, [math]\displaystyle{ \sinh(x) }[/math] is the hyperbolic sine function, [math]\displaystyle{ e }[/math] is natural exponential function and [math]\displaystyle{ x }[/math] is a real number.

• See: Hyperbolic Tangent Function, Complex Exponential Function, Hyperbolic Cosine Function.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Hyperbolic_function#Standard_analytic_expressions
  • QUOTE:(...) Hyperbolic sine:

[math]\displaystyle{ \sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x} = \frac {1 - e^{-2x}} {2e^{-x}} }[/math].

(...) Hyperbolic functions can be introduced via imaginary circular angles:

• Hyperbolic sine:

[math]\displaystyle{ \sinh x = -i \sin (i x) }[/math]

1999

• (Wolfram Mathworld , 1999) ⇒ http://mathworld.wolfram.com/HyperbolicSine.html
  • QUOTE: The hyperbolic sine is defined as

[math]\displaystyle{ \sinh z=\frac{1}{2}(e^z-e^{-z}) }[/math].

(...) Special values include [math]\displaystyle{ \sinh (0)=0, \sinh(\ln\phi)=1/2 }[/math] where [math]\displaystyle{ \phi }[/math] is the golden ratio. The value [math]\displaystyle{ \sinh(1)=1.17520119\dots }[/math] (...) has Engel expansion [math]\displaystyle{ 1, 6, 20, 42, 72, 110,\dots }[/math] (...) which has closed form [math]\displaystyle{ 2n(2n+1) \quad\textrm{for}\quad n\gt 1 }[/math]. The derivative is given by

[math]\displaystyle{ \frac{d\;\sinh z}{dz}= \cosh z }[/math]

where [math]\displaystyle{ \cosh z }[/math] is the hyperbolic cosine, and the indefinite integral by

[math]\displaystyle{ \int \sinh z\;dz=\cosh z+C }[/math]

where [math]\displaystyle{ C }[/math] is a constant of integration.

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