  We consider the quark anti-quark potential on the three sphere in planar N=4
SYM and the associated vacuum potential in the near BPS limit with L units of
R-charge. The associated Bremsstrahlung function B_L has been recently computed
analytically by means of the Thermodynamical Bethe Ansatz. We discuss it at
strong coupling by computing it at large but finite L. We provide strong
support to a special symmetry of the Bremsstrahlung function under the formal
discrete Z_2 symmetry L -> -1-L. In this context, it is the counterpart of the
reciprocity invariance discovered in the past in the spectrum of various gauge
invariant composite operators. The Z_2 symmetry has remarkable consequences in
the scaling limit where L is taken to be large with fixed ratio to the 't Hooft
coupling. This limit organizes in inverse powers of the coupling and resembles
the semiclassical expansion of the dual string theory which is indeed known to
capture the leading classical term. We show that the various higher-order
contributions to the Bremsstrahlung function obey several constraints and, in
particular, the next-to-leading term, formally associated with the string
one-loop correction, is completely determined by the classical contribution.
The large L limit at strong coupling is also discussed.
