(a) Show that, to the first order in the scattering length \(a\), the discontinuity in the specific heat \(C_{V}\) of an imperfect Bose gas at the transition temperature \(T_{c}\) is given by

\[\left(C_{V}\right)_{T=T_{c^{-}}}-\left(C_{V}\right)_{T=T_{c^{+}}}=N k \frac{9 a}{2 \lambda_{c}} \zeta(3 / 2)\]

while the discontinuity in the bulk modulus \(K\) is given by

\[(K)_{T=T_{c^{-}}}-(K)_{T=T_{c^{+}}}=-\frac{4 \pi a \hbar^{2}}{m v_{c}^{2}} .\]

(b) Examine the discontinuities in the quantities \(\left(\partial^{2} P / \partial T^{2}\right)_{v}\) and \(\left(\partial^{2} \mu / \partial T^{2}\right)_{v}\) as well, and show that your results are consistent with the thermodynamic relationship

\[C_{V}=V T\left(\frac{\partial^{2} P}{\partial T^{2}}\right)_{v}-N T\left(\frac{\partial^{2} \mu}{\partial T^{2}}\right)_{v}\]