General properties of Matsubara frequency summations By properties such as linearity, shifting, commutativity, etc. I was hoping to evaluate something like,
$$S_\eta = \dfrac{1}{\beta}\displaystyle\sum_{i\omega} \dfrac{i\omega-\xi_1}{[(i\omega-\xi_2)^2-\xi_3^2][(i\omega-\xi_4)^2-\xi_5^2]}$$
by using the results in this table. If not, would it be best to use partial fraction decomposition or is there another method?
 A: The result of your summation is
$$\frac{1}{2} \eta  \left(-\frac{\left(\xi _1-\xi _2+\xi
   _3\right) n_{\eta }\left(\xi _2-\xi _3\right)}{\xi _3
   \left(\xi _2^2-2 \left(\xi _3+\xi _4\right) \xi _2+\xi
   _3^2+\xi _4^2-\xi _5^2+2 \xi _3 \xi
   _4\right)}-\frac{\left(\xi _1-\xi _4+\xi _5\right)
   n_{\eta }\left(\xi _4-\xi _5\right)}{\left(\xi _2^2-2
   \left(\xi _4-\xi _5\right) \xi _2-\xi _3^2+\left(\xi
   _4-\xi _5\right)^2\right) \xi _5}-\frac{\left(-\xi
   _1+\xi _2+\xi _3\right) n_{\eta }\left(\xi _2+\xi
   _3\right)}{\xi _3 \left(\xi _2^2+2 \left(\xi _3-\xi
   _4\right) \xi _2+\xi _3^2+\xi _4^2-\xi _5^2-2 \xi _3 \xi
   _4\right)}-\frac{\left(-\xi _1+\xi _4+\xi _5\right)
   n_{\eta }\left(\xi _4+\xi _5\right)}{\xi _5 \left(\xi
   _2^2-2 \left(\xi _4+\xi _5\right) \xi _2-\xi
   _3^2+\left(\xi _4+\xi _5\right)^2\right)}\right)$$
The Matsubara frequency summation can be calculated analytically using the Mathematica package MatsubaraSum, which can be found at this Github repository: https://github.com/EverettYou/MatsubaraSum.
