convergence in the left half plane (Re z < 0), while controls the convergence in the right half plane (Re z > 0). Here is the Fermi–Dirac distribution function. In the application to Green's function calculation, g(z) always have the structure which diverges in the left half plane given 0 < τ < β. So as to control the convergence, the weighting function of the first type is always chosen . However, there is no need to control the convergence if the Matsubara summation does not diverge, in that case, any choice of the Matsubara weighting function will lead to identical results. Table of Matsubara frequency summations The following table contains for some simple rational functions g(z). The symbol η = ±1 is the statistical sign. [1] Since the summation does not converge, the result may differ upon different choice of the Matsubara weighting function. [2] (1 ↔ 2) denotes the same expression as the before but with index 1 and 2 interchanged. Applications in physics Zero temperature limit In this limit , the Matsubara frequency summation is equivalent to the integration of imaginary frequency over imaginary axis. Some of the integrals do not converge. They should be regularized by introducing the frequency cutoff , and then subtracting the divergent part (-dependent) from the integral before taking the limit of . For example, the free energy is obtained by the integral of logarithm, meaning that at zero temperature, the free energy simply relates to the internal energy below the chemical potential. Also the distribution function is obtained by the following integral which shows step function behavior at zero temperature. Green's function related Time domain Consider a function G(τ) defined on the imaginary time interval (0,β). It can be given in terms of Fourier series, where the frequency only takes discrete values spaced by 2/β. The particular choice of frequency depends on the boundary condition of the function G(τ). In physics, G(τ) stands for the imaginary time representation of Green's function It satisfies the periodic boundary condition G(τ+β)=G(τ) for a boson field. While for a fermion field the boundary condition is anti-periodic G(τ + β) = −G(τ). Given the Green's function G(iω) in the frequency domain, its imaginary time representation G(τ) can be evaluated by Matsubara frequency summation. Depending on the boson or fermion frequencies that is to be summed over, the resulting G(τ) can be different. To distinguish, define with