Let $a$ and $b$ be two real numbers. If $a > 0$ and $a ^ { 2 } - b ^ { 2 } = 1$ show that there exists a unique $\theta \in \mathbb { R }$ such that $a = \operatorname { ch } \theta$ and $b = \operatorname { sh } \theta$.
Let $k$ be a non-negative integer. We define the function $f _ { k }$ from the interval $[ - 1,1 ]$ to itself by $f _ { k } ( x ) = \cos ( k \arccos x )$, and $T_k$ is the polynomial of degree $k$ that identifies with $f_k$ on $[-1,1]$. We denote by arcosh the inverse function of the hyperbolic cosine, defined from $[ 1 , + \infty [$ to $[ 0 , + \infty [$. Show that $$\forall x \in ] - \infty , - 1 ] \quad T _ { k } ( x ) = ( - 1 ) ^ { k } \cosh ( k \operatorname { arcosh } ( - x ) ) .$$
For all $(t,\theta)\in\mathbb{R}_+\times[0,2\pi]$, define $$F(t,\theta) = \begin{pmatrix} \frac{1}{\sqrt{3}}\operatorname{sh}(t)\cos(\theta) \\ \frac{1}{\sqrt{3}}\operatorname{sh}(t)\sin(\theta) \\ \operatorname{ch}(t) \end{pmatrix}.$$ Calculate, for all $\theta\in[0,2\pi]$, the hyperbolic length of the path $$\begin{array}{rcl} \gamma:[0,b] & \rightarrow & \mathcal{H} \\ t & \mapsto & F(t,\theta). \end{array}$$
Let $F:[0,2\pi]\times\mathbb{R}_+\rightarrow\mathcal{H}$ be the map defined by $$F(t,\theta) = \begin{pmatrix} \frac{1}{\sqrt{3}}\operatorname{sh}(t)\cos(\theta) \\ \frac{1}{\sqrt{3}}\operatorname{sh}(t)\sin(\theta) \\ \operatorname{ch}(t) \end{pmatrix}.$$ For all $(\theta,\alpha)\in[0,2\pi]\times\mathbb{R}_+$ show that $$d\left(F(t,\theta), F\left(t,\theta+\alpha e^{-t}\right)\right) \underset{t\rightarrow+\infty}{\longrightarrow} \operatorname{arcch}\left(1+\frac{\alpha^2}{8}\right)$$ and that the convergence is uniform on every compact subset of $[0,2\pi]\times\mathbb{R}_+$.