  In this paper we study the expectation value of deformations of the circular
Wilson loop in ${\cal N}=4$ super Yang-Mills theory. The leading order
deformation, known as the Bremsstrahlung function, can be obtained exactly from
supersymmetric localization, so our focus is on deformations at higher orders.
We find simple expressions for the expectation values for generic deformations
at the quartic order at one-loop at weak coupling and at leading order at
strong coupling. We also present a very simple algorithm (not requiring
integration) to evaluate the two-loop result. We find that an exact symmetry of
the strong coupling sigma-model, known as the spectral-parameter independence,
is an approximate symmetry at weak coupling, modifying the expectation value
starting only at the sextic order in the deformation. Furthermore, we find very
simple patterns for how the spectral parameter can appear in the weak coupling
calculation, suggesting all-order structures.
