In thermal quantum field theory, the Matsubara frequency summation (named after Takeo Matsubara) is the summation over discrete imaginary frequencies. It takes the following form where is the inverse temperature and the frequencies are usually taken from either of the following two sets (with ): bosonic frequencies: fermionic frequencies: The summation will converge if tends to 0 in limit in a manner faster than . The summation over bosonic frequencies is denoted as (with ), while that over fermionic frequencies is denoted as (with ). is the statistical sign.