Show that the free energy \(A\) and the inertial density \(ho\) of a roton gas in mass motion are given by

\[
A(v)=A(0) \frac{\sinh x}{x}
\]

and

\[
ho(v)=ho(0) \frac{3(x \cosh x-\sinh x)}{x^{3}},
\]

where \(x=v p_{0} / k T\).