for example. In the heavy-light diquark calculation in Chapter 4 , more complicated functions involving dimensionless ratios of the external momentum Q 2 and the quark mass m occur. For instance, in the integral (2.67) the logarithms can be written as which has the appropriate branch cut structure. Using (2.74), we can define so that, for instance, Note the occurrence of the dilogarithm function in the integral (2.67). Using the definition of the dilogarithm function (B.18) and the properties of the logarithm above, the imaginary part can be shown to be Finally, we will consider the hypergeometric functions occurring in the heavy quarkonium hybrid calculations in Chapter 3. As mentioned previously, the hypergeometric function that occurs in (2.73) can be expressed in terms of inverse sine functions. Once this has been done, the imaginary part can be extracted using the identity where the logarithm can be dealt with as above. Using these techniques, closed form ex- pressions for the imaginary parts of the correlation functions in Chapters 3, 4 and 6 can be determined. 77 sin − $^{1}$( w ) = − i log $^{[}$iw + √ 1 − w 2 $^{]}$, (2.81) Im Li$_{2}$$^{[}$ 1 1 + m 2 Q 2 ] ∣ ∣ ∣ Q $^{2}$= te − $^{iπ}$,t>m 2 = iπ log$^{[}$1 − m 2 t ] . (2.80) Im log$^{[}$1 + Q 2 m 2 ] = − π , Im log 2 [ 1 + Q 2 m 2 ] = − 2 π log$^{[}$ t m 2 − 1$^{]}$ . (2.79) log$^{[}$1 + Q 2 m 2 ] ∣ ∣ Q $^{2}$= te − $^{iπ}$,t>m 2 ≡ log$^{[}$ t m 2 − 1$^{]}$ − iπ , (2.78) log [1 − z ] = − log$^{[}$1 + Q 2 m 2 ] , z = 1 1 + m 2 Q 2 . (2.77)