A Rayleigh wave, often the most visible component in the far‐field seismograms, is an important type of seismic‐wave motion associated with the Earth’s surface. In this study, we explore some of the general properties of the Rayleigh wave in a homogeneous elastic half‐space. Starting from the displacement expressed in the form of a wavenumber integral in the frequency domain, we extract the contribution from the pole in the complex wavenumber plane to obtain the excitation formulae of the Rayleigh wave by the residue theorem for complex integrals. Numerical results are compared with the full wavefield solutions to validate our solutions. By examining the analytical expressions obtained, we explore some basic properties of Rayleigh waves such as the particle motion and geometrical spreading. We also demonstrate that these properties of the Rayleigh wave excited by a point source are slightly different from but mostly consistent with the well‐known classical properties of plane Rayleigh waves.