content_image/402326.jpg content_image/402329.jpg content_image/402330.jpg content_image/402335.jpg content_image/402336.jpg content_image/402334.jpg content_image/402340.jpg content_image/402344.jpg content_image/402343.jpg
<img src='content_image/402326.jpg'>

The quantity Ω$_{0}$$_{,}$$_{0}$ in Eq. (A2) needs special attention. Consider the p = q = 0 term in the product for Ω$_{0}$$_{,}$$_{0}$, 

<img src='content_image/402329.jpg'>

Making use of (A3), we have for the last line of (A9) 

<img src='content_image/402330.jpg'>

According to (3.10),the critical temperature is at 

which can be rewritten as 

<img src='content_image/402335.jpg'>

<img src='content_image/402336.jpg'>

Thus, the p = q = 0 term in the product for Ω$_{00}$ is proportional to | θ − θ$_{c}$ | . The term going as the sign of that expression guarantees that the product goes as θ − θ$_{c}$ . It is a straightforward exercise to verify that all other factors in the product are free of singularities in the vicinity of the critical point. 

Now we focus on the sum over the variable q in Eq. (A8). There are two cases of interest: ν = 0 and ν = 1 / 2. We will analyze the case ν = 0 making use of the method outlined in Section III C. The generic sum of interest is 

which is equivalent to the integral 

## 1. Sum over the variable q 

<img src='content_image/402334.jpg'>

<img src='content_image/402340.jpg'>

The integral is around the contours shown in Fig. 4. This integral is evaluated by closing it around the two poles in the first term in the intregrand on the third line of (A14). We find after evaluating residues that the result of this integration is 

<img src='content_image/402344.jpg'>

Setting a/b + $^{√}$( a/b ) 2 − 1 = Γ and using the fact that a/b − $^{√}$( a/b ) 2 − 1 = ( a/b + $^{√}$( a/b ) 2 − 1) − $^{1}$, we can write (A15) as follows 

<img src='content_image/402343.jpg'>

