Use a virial expansion approach to determine the first few nontrivial order contributions to the pair correlation function \(g(r)\) in \(d\) dimensions. Show that the pair correlation function is of the form \(g(r)=e^{-\beta u(r)} y(r)\), where \(u(r)\) is the pair potential and \(y(r)\) is a smooth function of \(r\). Show that even for the case of hard-sphere interaction, \(y(r)\) and its first few derivatives are continuous.