such a function to the sum rule is given by which is independent of the continuum threshold s$_{0}$ . The following result is useful in order to calculate the Borel transform [97]: Note that Eq. (1.89) can be extended to cases where the denominator is raised to a higher power by differentiating with respect to t$_{0}$ . Functions that have a branch cut can be dealt with using the relationship between the Borel transform and the inverse Laplace transform (1.82). The contribution to the sum rule is given by R pole k ( τ ) = ˆ B τ [ $^{(}$− Q 2 ) k Π pole $^{(}$Q 2 ) ] , (1.88) ˆ B τ $^{[}$( − Q $^{2}$) k Q 2 + t$_{0}$$^{]}$ = t 2 k 0 e − t$_{0}$τ . (1.89) R branch k ( τ , s$_{0}$ ) = ˆ B τ [ $^{(}$− Q 2 ) k Π branch $^{(}$Q 2 ) ] − 1 π ∫ ∞ s$_{0}$ dt t k e − tτ ImΠ branch $^{(}$te − iπ ) , (1.90) Figure 1.8: Contour integral used to calculate the inverse Laplace transform in Eq. (1.91). The function has a branch cut on the interval Q 2 ∈ ( −∞ , − t$_{0}$ ], where t$_{0}$ is the hadronic threshold. 39