# 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:

 $\ds x$ $=$ $\ds a \cosh \theta$ $\ds y$ $=$ $\ds b \sinh \theta$