Schwarz conjugation of a spectral function ($F^{♯}(k)=\overline{F(\overline{k})}$)

This operation reflects a holomorphic function across the real axis while preserving holomorphic dependence on the new argument. A transform of real data with real kernel coefficients satisfies $F^{♯}=F$. In a polygonal global relation, complex rotation factors are conjugated as well; an explicit prefactor $i$ changes sign.