Could someone clear up the paper on casimir effect

[epislon58]

Hello,

could someone please explain to me what happens between (3.6) and (3.7). In specific, I don't understand how substitution results in l^2/pie^2 and also what do they mean integrating over radial angle. And the x next to the dx at (3.7), where did it come from.

http://aphyr.com/data/journals/113/comps.pdf

Thank you!

 

[epislon58]

anyone?

 

[atyy]

I didn't look that carefully but xdx is because x is a radial variable, so in the change of coordinates from Cartesian to polar, we usually get rdrdθ, where r is the Jacobian for the change of variables - they use x where I wrote r. The dθ doesn't appear because it has been integrated over already. Originally, they integrated over positive values of n_x and n_y, which are the original Cartesian coordinates. This is the first quadrant, ie. θ=0 to θ=∏/2.