A detailed analysis of a prolate hemispheroidal dielectric resonator antenna excited by a rectangular slot aperture is presented. The dyadic Green’s functions pertaining to a magnetic-current source located in a dielectric spheroid is derived and expressed in a form convenient for numerical computations. The dyadic Green’s functions are then employed to formulate the electromagnetic fields radiated by a prolate hemispheroidal dielectric resonator antenna. The electromagnetic far field is expressed, in a compact form, analytically. Furthermore, algorithms for computing the prolate spheroidal wave functions are developed and implemented using the software package MATLAB.