Definition:Hyperbolic Tangent/Real

Definition

Definition 1

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

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

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

Definition 2

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

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

$\forall x \in \R: \tanh x := \dfrac {\sinh x} {\cosh x}$

where:

$\sinh$ is the real hyperbolic sine

$\cosh$ is the real hyperbolic cosine

Also denoted as

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

Also see

• Equivalence of Definitions of Real Hyperbolic Tangent

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

• Definition:Real 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