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

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