The first term in Eq. (1.90) is an inverse Laplace transform This can be calculated using the residue theorem. For instance, consider the contour integral where the integration contour C is depicted in Fig. 1.8. By the residue theorem I = 0, and hence where I$_{L}$ , I$_{R}$ , I$_{top}$ , I$_{η}$ and I$_{bottom}$ denote the portion of contour C in Fig 1.8 parallel to the imaginary axis, the radial contour, the portion above the branch cut, the portion that circles the branch cut, and the portion below the branch cut, respectively. The exponential factor in Eq. (1.91) ensures that I$_{R}$ = 0 for R → ∞ and I$_{η}$ = 0 provided that Π $^{branch}$( − t$_{0}$ ) is regular. It can be shown that the remaining portions of the contour give where we have used the fact that Π $^{branch}$( Q $^{2}$) must satisfy Schwarz reflection. Inserting this into Eq. (1.90), we find In general, the field theoretic correlation function Π ( Q $^{2}$) will contain functions that have a branch cut, and in order to formulate the contribution of these to the sum rule we must evaluate the imaginary part of these functions below the branch cut. In principle, only the imaginary part of the correlation function is needed in order to use Eq. (1.95). However, there 40 R branch k ( τ , s$_{0}$ ) = 1 π s$_{0}$ ∫ t$_{0}$ dt t k e − tτ ImΠ branch $^{(}$te − iπ ) . (1.95) I$_{L}$ = 1 2 πi ∞ ∫ t$_{0}$ dt t k $^{[}$Π branch $^{(}$te − iπ ) − Π branch $^{(}$te iπ $^{)]}$e − tτ = 1 π ∞ ∫ t$_{0}$ dt t $^{k}$ImΠ $^{(}$te − iπ $^{)}$e − tτ , (1.94) I$_{L}$ = − I$_{R}$ − I$_{top}$ − I$_{η}$ − I$_{bottom}$ , (1.93) I = 1 2 πi ∮ C dQ 2 $^{(}$− Q 2 ) k Π branch $^{(}$Q 2 $^{)}$e Q $^{2}$τ , (1.92) L − 1 [ $^{(}$− Q 2 ) k Π branch $^{(}$Q 2 ) ; τ ] = 1 2 πi b + i ∞ ∫ b − i ∞ dQ 2 $^{(}$− Q 2 ) k Π branch $^{(}$Q 2 $^{)}$e Q $^{2}$τ . (1.91)