In this Letter, we analyze the reflection of cylindrical waves (CWs) at planar interfaces. We consider the reflected CW proposed in the literature as a spectral integral. We present a Laurent series expansion of the Fresnel coefficient convergent on the whole real axis and we use it to solve analytically the reflected-wave integral. We found a solution that involves both Bessel functions and Anger-Weber functions, i.e., solutions of both the homogeneous and inhomogeneous Bessel differential equations. We compare the analytical solution with the numerical results obtained with a quadrature formula presented in the literature. Moreover, we present a physical interpretation that connects our solution to the image principle.