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29) by the complex integral I D dp f . ip/ Á 2 i Z C1 i0C 1 Z C i0C dp f . ip/ 2 1Ci0C C1Ci0C dp f . 32) where the integration contour runs anti-clockwise around the real axis of the complex p-plane. 2 Evaluation of Thermal Sums and Their Low-Temperature Limit 23 The above result can be further simplified by substituting p ! term of Eq. 32), and noting that nB . ip/ D 1 exp. 33) This leads to the formula Z o dp n f . p/ C Œ f . p/ C f . ip/ 1 i0C 2 Z C1 i0C Z C1 dp dp f . p/ C Œ f . p/ C f . 34) where we returned to the real axis in the first term, made possible by the lack of singularities there.

Eq. 21) V Upon carrying out the contractions in Eq. e. the second one in Eq. 20), becomes clear: it cancels all disconnected contractions where all fields at point X are contracted with other fields at the same point. In other words, the combination in Eq. 20) amounts to taking into account only the connected contractions; this is the meaning of the subscript c in Eq. 12). This combinatorial effect is caused by the logarithm in Eq. , by going from the partition function to the free energy. 22) Inspecting the 2-point correlators in this result, we note that they either depend on X Y, or on neither X nor Y, the latter case corresponding to the contraction of fields at the same point.

102) We have gone one order higher in the middle expansion, because this function is multiplied by 1= in the result (cf. Eq. 112)). s/ is also to be viewed as a complex-valued function of a complex argument s. 1 Pequivalence 1 x nx e / D nD1 e , and using the definition of the €-function in Eq. 96). 104) and then substituting integration variables through x ! s/ > 0, s ¤ 1. Even though the integral here clearly diverges at s ! s/ has a pole only at s D 1. Its values at even arguments are “easy”; in fact, at even negative integers, Eq.

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