Obtain an asymptotic expression for the quantity \(\ln g(E)\) for a system of \(N\) quantum-mechanical harmonic oscillators by using the inversion formula (3.4.7) and the partition function (3.8.15). Hence show that

\[
\frac{S}{N k}=\left(\frac{E}{N \hbar \omega}+\frac{1}{2}ight) \ln \left(\frac{E}{N \hbar \omega}+\frac{1}{2}ight)-\left(\frac{E}{N \hbar \omega}-\frac{1}{2}ight) \ln \left(\frac{E}{N \hbar \omega}-\frac{1}{2}ight)
\]