Fraunhofer holography is re-examined, correcting an error that has existed in the literature concerning the equation that describes the amplitude transmittance of a Fraunhofer hologram. In addition, the cases involving one and two-dimensional apertures are examined in detail, showing that a 45° phase shift exists between them. The original analysis is then extended to include a detailed description of the reconstructed image. This is accomplished by treating the hologram as a new diffracting aperture of finite extent, yielding, for the first time, a complete analytical description of the reconstructed field associated with a Fraunhofer hologram.