Definition:Hyperbolic Function

Definition

There are six 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

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): hyperbolic function
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbolic functions
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbolic functions
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): hyperbolic function