Existing formulations of the three-dimensional (3-D) diffraction pattern of spherical waves that is produced by a circular aperture are reviewed in the context of 3-D serial-sectioning microscopy. A new formulation for off-axis focal points is introduced that has the desirable properties of increased accuracy for larger field angles, invariance to shifts of the focal point about spheres of constant radius when the detection point is on the sphere for both intensity and amplitude fields, and invariance to shifts in three transformed coordinates for intensity fields. Finally, calculated intensity fields for both on-axis and off-axis focal points are included to illustrate the proposal that the classical 3-D diffraction patterns that have been used as analytical models in 3-D serial-sectioning fluorescence microscopy may not be accurate enough for this application.