content_image/416201.jpg content_image/416202.jpg content_image/416203.jpg content_image/416204.jpg content_image/416200.jpg content_image/416205.jpg content_image/416206.jpg content_image/416208.jpg content_image/416209.jpg content_image/416210.jpg content_image/416212.jpg content_image/416213.jpg content_image/416207.jpg content_image/416211.jpg content_image/416214.jpg content_image/416215.jpg content_image/416216.jpg
and where we have defined the dimensionless tempera- ture 

Similarly to the case at zero temperature this simple ex- pression is in need of regularization in order to give nu- merical meaning since it is formally divergent. 

## III. REGULARIZATION OF THE FREE ENERGY EXPRESSION 

We here follow a scheme closely reminiscent of that of Milton and DeRaad [8], and particularly Milton, Nesterenko and Nesterenko [10] (cf. also Appendix A of Ref. [34]). 

As follows from the uniform asymptotic expansion of modified cylindrical Bessel functions, e.g. § 9.7 of Ref. [39], the logarithmic factor in the integrand of (2.7) has the asymptotic behavior 

<img src='content_image/416201.jpg'>

<img src='content_image/416202.jpg'>

To see how this behavior gives rise to a formal diver- gence, consider the case of large m for which 

<img src='content_image/416203.jpg'>

where we substituted y 2 = x 2 − j $^{2}$τ2 and defined the shorthand 

The three terms of (3.2b) correspond to the three terms of the integrand of (3.2a). All of the terms of (3.2b) clearly diverge when summed over j and m . 

The first step in regularization is to add and subtract the asymptotic behavior (3.1) in the form (3.2a) 

where we define the energy with the leading asymptotic 

<img src='content_image/416204.jpg'>

<img src='content_image/416200.jpg'>

<img src='content_image/416205.jpg'>

<img src='content_image/416206.jpg'>

term subtracted, 

<img src='content_image/416208.jpg'>

<img src='content_image/416209.jpg'>

and the additional, non-regularized energy 

<img src='content_image/416210.jpg'>

wherein 

$^{To regularize}$E ∞ we introduce the small quantity s and write 

<img src='content_image/416212.jpg'>

where as in Eq. (3.2b) we have used the evaluation, 

<img src='content_image/416213.jpg'>

which is valid for − 1 < s < 5, so it may be used for s near 0, near 2, or near 4. We use the relations [ ∂$_{τ}$ = ∂/∂τ ] 

to write 

<img src='content_image/416207.jpg'>

<img src='content_image/416211.jpg'>

<img src='content_image/416214.jpg'>

<img src='content_image/416215.jpg'>

<img src='content_image/416216.jpg'>

