Definition:Hyperbolic Function

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Definition

There are six basic hyperbolic functions, as follows:


Hyperbolic Sine

The hyperbolic sine function is defined on the complex numbers as:

$\sinh: \C \to \C$:
$\forall z \in \C: \sinh z := \dfrac {e^z - e^{-z} } 2$


Hyperbolic Cosine

The hyperbolic cosine function is defined on the complex numbers as:

$\cosh: \C \to \C$:
$\forall z \in \C: \cosh z := \dfrac {e^z + e^{-z} } 2$


Hyperbolic Tangent

The hyperbolic tangent function is defined on the complex numbers as:

$\tanh: X \to \C$:
$\forall z \in X: \tanh z := \dfrac {e^z - e^{-z} } {e^z + e^{-z} }$

where:

$X = \set {z : z \in \C, \ e^z + e^{-z} \ne 0}$


Hyperbolic Cotangent

The hyperbolic cotangent function is defined on the complex numbers as:

$\coth: X \to \C$:
$\forall z \in X: \coth z := \dfrac {e^z + e^{-z} } {e^z - e^{-z}}$

where:

$X = \set {z : z \in \C, \ e^z - e^{-z} \ne 0}$


Hyperbolic Secant

The hyperbolic secant function is defined on the complex numbers as:

$\sech: X \to \C$:
$\forall z \in X: \sech z := \dfrac 2 {e^z + e^{-z} }$

where:

$X = \set {z: z \in \C, \ e^z + e^{-z} \ne 0}$


Hyperbolic Cosecant

The hyperbolic cosecant function is defined on the complex numbers as:

$\csch: X \to \C$:
$\forall z \in X: \csch z := \dfrac 2 {e^z - e^{-z} }$

where:

$X = \set {z: z \in \C, \ e^z - e^{-z} \ne 0}$


Also see

  • Results about hyperbolic functions can be found here.


Historical Note

The hyperbolic functions are so called because of their ability to be used to generate the parametric form of the equation of the hyperbola:

\(\ds x\) \(=\) \(\ds a \cosh \theta\)
\(\ds y\) \(=\) \(\ds b \sinh \theta\)


Sources

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