Hyperbolic functions/R/Introduction/Section

The hyperbolic functions.

Definition

The function defined for $x\in ℝ$ by

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

is called hyperbolic sine.

Definition

The function defined for $x\in ℝ$ by

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

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.

$◻$

Lemma

The function hyperbolic sine is strictly increasing, and the function hyperbolic cosine

is strictly decreasing on ${ℝ}_{\le 0}$ and strictly increasing on ${ℝ}_{\ge 0}$.

Proof

See exercise and exercise.

$◻$

Definition

The function

$ℝ⟶ℝ,x⟼\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}},$

is called hyperbolic tangent.