Hyperbolic Cosine Function

A Hyperbolic Cosine Function is a hyperbolic function based on the cosine function.

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

[math]\displaystyle{ \cosh x = \cos (i x)= \frac {e^x + e^{-x}} {2} }[/math]

where [math]\displaystyle{ \cos(x) }[/math] is the cosine function, [math]\displaystyle{ \cosh(x) }[/math] is the hyperbolic cosine 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 Sine Function.

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References

2015

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

[math]\displaystyle{ \cosh 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 (...):

[math]\displaystyle{ \cosh x = \cos (i x) }[/math]

1999

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

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

(...)This function describes the shape of a hanging cable, known as the catenary(...). Special values include [math]\displaystyle{ \cosh(0) =1, \; \cosh(\ln\phi) =1/2\sqrt{5} }[/math], where [math]\displaystyle{ \phi }[/math] is the golden ratio. The derivative is given by

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

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

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

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

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