  We explicitly compute the semi-global symplectic invariants near the
focus-focus point of the spherical pendulum. A modified Birkhoff normal form
procedure is presented to compute the expansion of the Hamiltonian near the
unstable equilibrium point in Eliasson-variables. Combining this with explicit
formulas for the action we find the semi-global symplectic invariants near the
focus-focus point introduced by Vu Ngoc 2003. We also show that the Birkhoff
normal form is the inverse of a complete elliptic integral over a vanishing
cycle. To our knowledge this is the first time that semi-global symplectic
invariants near a focus-focus point have been computed explicitly. We close
with some remarks about the pendulum, for which the invariants can be related
to theta functions in a beautiful way.
