Hyperbolic functions/R/Introduction/Section

The hyperbolic functions.

Definition

The function defined for ${}x\in \mathbb {R}$ by

${}\sinh x:={\frac {1}{2}}{\left(e^{x}-e^{-x}\right)}\,,$

is called hyperbolic sine.

Definition

The function defined for ${}x\in \mathbb {R}$ by

${}\cosh x:={\frac {1}{2}}{\left(e^{x}+e^{-x}\right)}\,,$

is called hyperbolic cosine.

The cosine hyperbolicus ${}a\cosh x/a$ (with parameter ${}a$) describes a so-called catenary, that is, the curve of a hanging chain.

Lemma

The functions hyperbolic sine and hyperbolic cosine

have the following properties.

1. ${}\cosh x+\sinh x=e^{x}\,.$

2. ${}\cosh x-\sinh x=e^{-x}\,.$

3. ${}(\cosh x)^{2}-(\sinh x)^{2}=1\,.$

Proof

See exercise.

$\Box$

Lemma

The function hyperbolic sine is strictly increasing, and the function hyperbolic cosine is strictly decreasing on ${}\mathbb {R} _{\leq 0}$ and strictly increasing on ${}\mathbb {R} _{\geq 0}$.

Proof

See exercise and exercise.

$\Box$

Definition

The function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}},$

is called hyperbolic tangent.