content_image/306215.jpg content_image/306216.jpg content_image/306219.jpg content_image/306221.jpg content_image/306220.jpg content_image/306217.jpg content_image/306218.jpg
where µ is the chemical potential. Therefore we obtain (with τ =0 as µ =0 ) 

Substituting this into Eq. (A.13) gives 

<img src='content_image/306215.jpg'>

<img src='content_image/306216.jpg'>

Now, we can obtain Eq. (A.17) from Eq. (A.12) by simply changing the integration variable as 

<img src='content_image/306219.jpg'>

<img src='content_image/306221.jpg'>

<img src='content_image/306220.jpg'>

We can make progress in trying to solve the integral of J (1) − as follows. J ( − 1) can be written as 

<img src='content_image/306217.jpg'>

where K is the modified Bessel function of the second kind. There exists a recurrence relation 

<img src='content_image/306218.jpg'>

