  Mathieu's equation has many applications throughout theoretical physics. It
is especially important to the theory of Josephson junctions, where it is
equivalent to Schrodinger's equation. Mathieu's equation can be easily solved
numerically, however there exists no closed-form analytic solution. Here we
collect various approximations which appear throughout the physics and
mathematics literature and examine their accuracy and regimes of applicability.
Particular attention is paid to quantities relevant to the physics of Josephson
junctions, but the arguments and notation are kept general so as to be of use
to the broader physics community.
