Category:Definitions/Hyperbolic Functions

This category contains definitions related to Hyperbolic Functions.
Related results can be found in Category:Hyperbolic Functions.

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:

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