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\newcommand{\un}{\underline}
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\begin{document}

{\bf Question}

For integers $m\ge 0$, consider the paths $g_m: [0,1]\rightarrow
{\bf H}$ given by
\[ g_m(t) = t + (t^{3m} + 1) i. \]
Write down the integral giving the hyperbolic length of the curve
$g_m([0,1])$ in ${\bf H}$.   Evaluate it if you can.

\medskip
\noindent By considering what the curves $g_m([0,1])$ look like in
${\bf H}$ as $m\rightarrow \infty$, determine the putative limit
of the hyperbolic lengths ${\rm length}_{\bf H} (g_m([0,1]))$ as
$m\rightarrow\infty$.
\medskip

{\bf Answer}

Im$(g_m)=t^{3m}+1$

$|g'_m(t)|=|1+3mt^{3m-1}i|=\sqrt{1+9m^2t^{6m-2}}$

\bigskip

So length$_{\bf H}(g_m)=\ds\int_0^1
\ds\frac{1}{t^{3m}+1}\sqrt(1+9m^2t^{6m-2} \,dt.$

\bigskip

\un{$m=0$} length$_{\bf H}(g_0)=\ds\int_0^1
\ds\frac{1}{1+1}\sqrt{1+0} \,dt=\ds\frac{1}{2}$

\bigskip

\un{$m=1$} length$_{\bf H}(g_1)=\ds\int_0^1
\ds\frac{1}{t^3+1}\sqrt{1+9t^4} \,dt$

\bigskip

and others that I don't know how to evaluate.

\bigskip

But, as $m \to \infty$, $g+m([0,1])$ approaches the union of the
horizontal Euclidean line segment from $i$ to $i+1$ and the
vertical line segment from $1+i$ to $1+2i$.

\bigskip

So, this horizontal Euclidean line segment is parametrized by
$f:[0,1] \longrightarrow {\bf H},\ f(t)=t+i$ and so length$_{\bf
H}(f)= \ds\int_0^1 \,dt=1$

\bigskip

and the vertical line segment has length$_{\bf H}=\ln(2)=d_{\bf
H}(1+i,1+2i)$.

So \un{length$_{\bf H}(g_m) \longrightarrow 1+\ln(2)$} as $m \to
\infty$.
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