  A cylindrical wave expansion method is developed to obtain the scattering
field for an ideal two-dimensional cylindrical invisibility cloak. A near-ideal
model of the invisibility cloak is set up to solve the boundary problem at the
inner boundary of the cloak shell. We confirm that a cloak with the ideal
material parameters is a perfect invisibility cloak by systematically studying
the change of the scattering coefficients from the near-ideal case to the ideal
one. However, due to the slow convergence of the zero$^{th}$ order scattering
coefficients, a tiny perturbation on the cloak would induce a noticeable field
scattering and penetration.
