The modified Watson transform is applied to the Mie series expression of the electromagnetic field scattered by a high frequency plane wave incident on an infinitely long double negative cylinder. The Debye expansion is applied to the Mie series coefficients to obtain a physical insight into the scattering mechanisms and achieve an efficient approach for the computation of the scattered field. The first two terms of the Debye series are computed using the residue series in the geometrical shadow regions and using the steepest descent method in the geometrically lit regions. It is observed that the results obtained from the series and from the modified Watson transform are in good agreement. The angular boundaries for the geometrically lit and the geometrical shadow regions of the double negative cylinder corresponding to the first two terms of the Debye series are determined. These are compared with the corresponding angular boundaries for a double positive cylinder. It is observed that the spatial extent of the geometrical shadow of the double negative cylinder corresponding to the second term of the Debye series is very small compared to that of the double positive cylinder due to the negative refraction in the double negative cylinder when the magnitude of the refractive index n is greater than root 2.