Laplace transform grand partition function

There is a standard procedure to proceed from (2.5.21) to obtain the Laplace transform of the grand partition function with respect to the length variable i.e. 

By choosing a specific ordering of the species then applying the convolution theorem, the Laplace transform of the grand partition function can be rewritten as... 

Normally, to recover the grand partition function, one needs to take the inverse Laplace transform of (2.5.26). However, in our case, we can view the Laplace transform of S(T,L,A) as the generalized partition function, and we can obtain all the thermodynamic quantities of the system from the eigenvalues of the matrix... 

In order to use the Laplace transform of the grand partition function in (2.5.26), y must be equal to 1 hence, the sum on the right-hand side of (2.5.26) diverges. This leads to the implicit equation of state of the form... 

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