  In this paper, we develop a new method of computing three-point functions in
the SU(2) sector of the $\mathcal{N}=4$ super Yang-Mills theory in the
semi-classical regime at weak coupling, which closely parallels the strong
coupling analysis. The structure threading two disparate regimes is the
so-called monodromy relation, an identity connecting the three-point functions
with and without the insertion of the monodromy matrix. We shall show that this
relation can be put to use directly for the semi-classical regime, where the
dynamics is governed by the classical Landau-Lifshitz sigma model.
Specifically, it reduces the problem to a set of functional equations, which
can be solved once the analyticity in the spectral parameter space is
specified. To determine the analyticity, we develop a new universal logic
applicable at both weak and strong couplings. As a result, compact
semi-classical formulas are obtained for a general class of three-point
functions at weak coupling including the ones whose semi-classical behaviors
were not known before. In addition, the new analyticity argument applied to the
strong coupling analysis leads to a modification of the integration contour,
producing the results consistent with the recent hexagon bootstrap approach.
This modification also makes the Frolov-Tseytlin limit perfectly agree with the
weak coupling form.
