+ {1-\exp(-\beta\hbar\omega_s({\bf k}))}\nonumber\\
+ &=&-\frac{1}{V}\frac{\partial}{\partial \beta} \sum_{{\bf k}s} ln
+ \frac{\exp(-\beta\hbar\omega_s({\bf k})/2)}
+ {1-\exp(-\beta\hbar\omega_s({\bf k}))}\nonumber\\
+ &=&-\frac{1}{V}\sum_{{\bf k}s}
+ \frac{1-\exp(-\beta\hbar\omega_s({\bf k}))}
+ {\exp(-\beta\hbar\omega_s({\bf k})/2)}\nonumber\\
+ &&\times
+ \frac{(1-e^{-\beta\hbar\omega_s({\bf k})})
+ e^{-\beta\hbar\omega_s({\bf k})/2}(-\hbar\omega_s({\bf k})/2)+
+ e^{-\beta\hbar\omega_s({\bf k})/2}
+ e^{-\beta\hbar\omega_s({\bf k})}(-\hbar\omega_s({\bf k}))}
+ {(1-e^{-\beta\hbar\omega_s({\bf k})})^2}\nonumber\\
+ &=&-\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
+ \frac{e^{-\beta\hbar\omega_s({\bf k})}-
+ \frac{1}{2}(1-e^{-\beta\hbar\omega_s({\bf k})})}
+ {1-e^{-\beta\hbar\omega_s({\bf k})}}\nonumber\\
+ &=&-\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
+ \frac{\frac{1}{2}e^{-\beta\hbar\omega_s({\bf k})}-\frac{1}{2}}
+ {1-e^{-\beta\hbar\omega_s({\bf k})}}
+ =\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})\frac{1}{2}
+ \frac{1+e^{\beta\hbar\omega_s({\bf k})}}
+ {e^{\beta\hbar\omega_s({\bf k})}-1}\nonumber\\
+ &=&\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})\frac{1}{2}
+ \frac{1+e^{\beta\hbar\omega_s({\bf k})}}
+ {e^{\beta\hbar\omega_s({\bf k})}-1}
+ =\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})\frac{1}{2}
+ \frac{2+e^{\beta\hbar\omega_s({\bf k})}-1}
+ {e^{\beta\hbar\omega_s({\bf k})}-1}\nonumber\\
+ &=&\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
+ (\frac{1}{e^{\beta\hbar\omega_s({\bf k})}-1}
+ +\frac{e^{\beta\hbar\omega_s({\bf k})}-1}
+ {2(e^{\beta\hbar\omega_s({\bf k})}-1)})
+ =\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
+ (\underbrace{\frac{1}{e^{\beta\hbar\omega_s({\bf k})}-1}}_{n_s({\bf k})}
+ +\frac{1}{2})\nonumber