$$\begin{align} I = \int \mathrm{d}^3k \left[ \frac{1}{(\vec{k} - \vec{k}_0)^{2}} + \frac{1}{(\vec{k} + \vec{k}_0)^{2}} - \frac{2}{k^2} \right] = 0 \end{align}$$

Mahtematica 可以直接得到

Integrate[k^2 (1/(k^2+k0^2- 2 k k0 x)-2/k^2+1/(k^2+k0^2+2 k k0 x)),{x,-1,1},{k,0, \[Infinity]}]

太失败了, 是一开始想复杂, 根本没有奇异性...

Reference

  • Tan, S. Energetics of a strongly correlated Fermi gas. Annals of Physics 323, 2952–2970 (2008).