  We describe a method for calculating action-angle variables in axisymmetric
galactic potentials using Birkhoff normalization, a technique from Hamiltonian
perturbation theory. An advantageous feature of this method is that it yields
explicit series expressions for both the forward and inverse transformations
between the action-angle variables and position-velocity data. It also provides
explicit expressions for the Hamiltonian and dynamical frequencies as functions
of the action variables. We test this method by examining orbits in a
Miyamoto-Nagai model potential and compare it to the popular St\"ackel
approximation method. When vertical actions are not too large, the Birkhoff
normalization method achieves fractional errors smaller than a part in $10^{3}$
and outperforms the St\"ackel approximation. We also show that the range over
which Birkhoff normalization provides accurate results can be extended by
constructing Pad\'e approximants from the perturbative series expressions
developed with the method. Numerical routines in Python for carrying out the
Birkhoff normalization procedure are made available.
