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We derive the first $\epsilon_2$-correction to the instanton partitionfunctions of $\mathcal{N}=2$ Super Yang-Mills (SYM) in four dimensions in theNekrasov-Shatashvili limit $\epsilon_2\rightarrow 0$. In the latter we recallthe emergence of the famous Thermodynamic Bethe Ansatz-like equation which hasbeen found by Mayer expansion techniques. Here we combine efficiently these tofield theory arguments. In a nutshell, we find natural and resolutive theintroduction of a new operator $\nabla$ that distinguishes the singularitieswithin and outside the integration contour of the partition function.

Finite $\epsilon_2$-corrections to the $\mathcal{N}=2$ SYM prepotential