We saw that the E(N,V,S) and A(N,V,T) could be related by a
Legendre transformation. The partition functions 
and
Q(N,V,T) can be related by a Laplace transform. Recall that the
Laplace transform 
of a function f(x) is given by
Let us compute the Laplace transform of with respect to E:
Using the 
-function to do the integral over E:
By identifying 
, we see that the Laplace transform
of the microcanonical partition function gives the canonical partition function
Q(N,V,T).