Topics in Statistic Mechanics

Adv. Sta. Phy. Homework 5 XY Model Li,Zimeng PB06203182
Here r=i-j, k'=-k (see previous proof),and the last part has changed rectangular
coordinates to polar coordinates,
), and is just the momentum part of the lattice constant a, and
therefore we can approximate ,
Noticing
We integrate the above equation by , and the second part
Therefore,
(3.2.1)
as , so (3.2.2),
as therefore the largest part of equ (3.2.2) is , and we can
approximate the lower limit of (3.2.1) by
so