Definition:Hyperbolic Tangent

Definition

The hyperbolic tangent is one of the hyperbolic functions:

Definition 1

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:

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

Definition 2

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

$\tanh: X \to \C$:

$\forall z \in X: \tanh z := \dfrac {\sinh z} {\cosh z}$

where:

$\sinh$ is the hyperbolic sine

$\cosh$ is the hyperbolic cosine

$X = \set {z : z \in \C, \ \cosh z \ne 0}$

Definition 3

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

$\tanh: X \to \C$:

$\forall z \in X: \tanh z := \dfrac {e^{2 z} - 1} {e^{2 z} + 1}$

where:

$X = \set {z: z \in \C, \ e^{2 z} + 1 \ne 0}$

Real Hyperbolic Tangent

On the real numbers it is defined similarly.

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

$\tanh: \R \to \R$:

$\forall x \in \R: \tanh x := \dfrac {e^z - e^{-x} } {e^z + e^{-x} }$

Also denoted as

The notation $\operatorname {th} z$ can also be found for hyperbolic tangent.

Also see

• Equivalence of Definitions of Hyperbolic Tangent

• Definition:Hyperbolic Sine
• Definition:Hyperbolic Cosine
• Definition:Hyperbolic Cotangent
• Definition:Hyperbolic Secant
• Definition:Hyperbolic Cosecant

• Definition:Inverse Hyperbolic Tangent

• Results about the hyperbolic tangent function can be found here.

Linguistic Note

The usual symbol tanh for hyperbolic tangent is awkward to pronounce.

Some pedagogues say it as tansh, and some as than (where the th is voiceless as in thin, for example).

Others prefer the mouthful which is hyperbolic tan.

Sources

• Weisstein, Eric W. "Hyperbolic Tangent." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicTangent.html