Anti-reflection(AR), a well-known technique of reducing unwanted reflections by applying an impedance matching layer, works for a specific wavelength and require the coating layer to be a quarter wavelength thick. A broadband operation of AR, however, is not fully understood except for the trial and error method. Here, we present a systematic analytic method of AR without the restriction of wavelength or thickness, i.e. achieving a perfect AR. Specifically, we find analytic permittivity and permeability profiles that remove any given impedance mismatch at the interface between two different dielectrics in a frequency independent way. Ultra-thin AR coating is also shown to be possible and confirmed experimentally with the l/25-wavelength thick AR coating layer made of metamaterials. We apply the concept of ultrathin double layer AR to the transparent conducting electrode, which we demonstrate by fabricating a low reflective dielectric/metal-layered electrode that provides significant electrical conductivity and light transparency.