Note that τ is restricted in the principal interval (0,β). The boundary condition can be used to extend G(τ) out of the principal interval. Some frequently used results are concluded in the following table. Operator switching effect The small imaginary time plays a critical role here. The order of the operators will change if the small imaginary time changes sign. Distribution function The evaluation of distribution function becomes tricky because of the discontinuity of Green's function G(τ) at τ = 0. To evaluate the summation both choices of the weighting function are acceptable, but the results are different. This can be understood if we push G(τ) away from τ = 0 a little bit, then to control the convergence, we must take as the weighting function for , and for . Bosons Fermions Free energy Bosons Fermions Diagram evaluations Frequently encountered diagrams are evaluated here with the single mode setting. Multiple mode problems can be approached by a spectral function integral. Fermion self energy Particle-hole bubble Particle-particle bubble Appendix: Properties of distribution functions Distribution functions The general notation stands for either Bose (η = +1) or Fermi (η = −1) distribution function If necessary, the specific notations nB and nF are used to indicate Bose and Fermi distribution functions respectively Relation to hyperbolic functions The Bose distribution function is related to hyperbolic cotangent function by The Fermi distribution function is related to hyperbolic tangent function by Parity Both distribution functions do not have definite parity, Another formula is in terms of the function However their derivatives have definite parity. Bose–Fermi transmutation Bose and Fermi distribution functions transmute under a shift of the variable by the fermionic frequency, However shifting by bosonic frequencies does not make any difference. Derivatives First order In terms of product: In the zero temperature limit: Second order Formula of difference Case a = 0 Case a → 0 Case b → 0 The function cη Definition: For Bose and Fermi type: Relation to hyperbolic functions It is obvious that is positive definite. To avoid overflow in the numerical calculation, the tanh and coth functions are used Case a = 0 Case b = 0 Low temperature limit For a = 0: For b = 0: In general, See also Imaginary time Thermal quantum field theory External links Agustin Nieto: Evaluating Sums over the Matsubara Frequencies. arXiv:hep-ph/9311210 Github repository: MatsubaraSum A Mathematica package for Matsubara frequency summation.