  The amplitude dependent tune shifts from either space charge or beam-beam
interactions are calculated analytically with the inclusion of synchrotron
oscillations and multiple interactions around the ring. Simpler formulae are
derived under limits of bunches longer than the transverse sizes, equal and
unequal transverse sizes etc. This is used to derive semi-analytical forms for
the density distribution of the tune shifts. The tune spread and the density
distribution are needed to understand beam decoherence or Landau damping with
either interaction. The tune footprints due to space charge in IOTA are
simulated using pyorbit and found to be in good agreement with the theoretical
predictions.
