  By using extended bosonic coherent states, the solution to the
Jaynes-Cummings model without the rotating-wave approximation can be mapped to
that of a polynomial equation with a single variable. The solutions to this
polynomial equation can give all eigenvalues and eigenfunctions of this model
with all values of the coupling strength and the detuning exactly, which can be
readily applied to recent circuit quantum electrodynamic systems operating in
the ultra-strong coupling regime.
