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The problem is to evaluate $\Gamma[\phi]$ at fixed $\phi$ given an expansion of $W[J]$ in powers of $\hbar$, where $\Gamma[\phi]$ is the Legendre transform of $W[J]$. By definition, $$ \Gamma[\phi]=\sup_{J}\Big[\phi\cdot J-W[J]\Big]. $$ Suppose that the expression in the RHS attains its maximum at some $\hat{J}$. Then formally, we can expand $\phi\cdot ...

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