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I don't understand how some terms are derived.

How did the last term of 3-47 originate?

How did 3-49 get so many terms from just one term in 3-41?

Why integrate from V to infinite? That is not intuitive.

Thas a functions are unusual because the absolute values of U,H,S cannot be computed. However, the analytical functions are needed to derive the expression for chemical potential?