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 zeroth order scattering coefficients, a tiny perturbation on the cloak would

induce a noticeable field scattering and penetration.