5.4. Generating function of dimers on a square lattice Consider the function F(x,y) = 1 (2)2 ...

5.4. Generating function of dimers on a square lattice Consider the function F(x,y) = 1

(2π)2

 2π

0

 2π

0 dβ1 dβ2 log [x+y−xcosβ1 −ycosβ2] .

Its value at x = y =, i.e.F(2,2), provides the solution to the problem of the dimer covering of a square lattice, eqn. (5.2.14).

a. Prove that F(x,0) = log x

2

.

b. Show that it holds the identity

∂F

∂x

= 2

πx arctan x y

.

c. Expanding in power series the term arctan!xy and integrating term by term, show that F(2,2) = 4G

π

where G is the Catalan constant.