Finally, expanding the remaining functions we find Note that the hypergeometric function can be expressed in terms of inverse sine functions. However, there are several reasons for leaving the result in terms of a hypergeometric func- tion. First, all hypergeometric functions with argument x have a branch cut on x ∈ [1 , ∞ ), therefore it is clear that the result (2.73) has appropriate branch cut structure, namely a branch cut on q 2 ∈ [4 m $^{2}$, ∞ ). However, when the result is expressed in terms of inverse sine functions, this branch cut structure is obscured. Second, the result is very compact. Apart from the hypergeometric function, all terms in (2.73) are dispersion relation subtraction constants. The methods discussed so far in this chapter are sufficient to calculate the correlation func- tions that are studied in Chapters 3, 4 and 6. However, it is the singularities of the Wilson coefficients in the complex Euclidean momentum plane that are of interest in QSR analyses. Typically, these singularities appear as isolated poles or as branch cuts. Fig. 2.7 shows a typical branch cut singularity. The branch point corresponds to the hadronic threshold, which is related to the total mass of the hadronic constituents. For instance, in the heavy quarkonium hybrid calculations in Chapter 3, the currents used contain two identical heavy quarks so that t$_{0}$ = 4 m $^{2}$. In calculations that involve light quarks we work in the chiral limit, ignoring the light quark mass. Accordingly, t$_{0}$ = m 2 in the heavy-light diquark calculation in Chapter 4, while in the glueball quark meson mixing calculation in Chapter 6, t$_{0}$ = 0. Recall that in QSR analyses 75 2.7 Analytic Continuation J ( d ; 1 , m ; 1 , m ; 1 , 0) = m 2 256 π 4 [ − 1 6$^{(}$ 42 + π 2 + 3 w + 12 log$^{[}$ m 2 µ 2 ][ w − 3 + log$^{[}$ m 2 µ 2 ]]) − 1 ϵ 2 + 1 ϵ ( 3 − w − 2 log$^{[}$ m 2 µ 2 ]) + 4 w 2 9 $_{3}$F$_{2}$$^{[}$ 1 , 1 , 2 ; 4 , 5 2 ; w$^{]}$$^{]}$ . (2.73)