  Very little previous literature has considered the *exact* solution to
Maxwell's equations for an infinite ideal cylindrical solenoid with an
arbitrary time-dependent azimuthal surface current $K(t) \hat{\bf \phi}$. Most
of the previous literature has focused on special cases and has approached the
problem by calculating the magnetic vector potential ${\bf A}$, which requires
performing some very complicated surface integrals over the cylinder. In this
article, we take a simpler approach and directly tackle Maxwell's equations
without ever invoking a vector potential. The high symmetry of the geometry
allows us to reduce Maxwell's equations to just two coupled partial
differential equations for two functions of two real variables, which can be
readily solved numerically. We find the general analytic solution to these PDEs
and derive the Green's functions for the electromagnetic fields, which allow us
to calculate the fields directly from the surface current $K(t)$. We also
briefly discuss a family of exact formal solutions that (the author believes)
has not appeared in the previous literature because it corresponds to a current
$K(t)$ that does not have a Fourier transform.
