Abstract: This paper deeply studies both zero-order angle prolate spheroidal wave and prolate spheroidal wave functions, and gives differential equations for describing both of them. It also gives the relations between the Spherical coordinates and the prolate spheroidal coordinates for Helmholtz equation. This paper provides an analysis method for the integral equation of the prolate spheroidal wave functionderived from the optimal design for signal waveforms, which solves the optimal waveform design. This paper also tries to explain the reason for the name of the prolate spheroidal wave function.