The computation of the two-dimensional harmonic spatial-domain Green's function at the surface of a piezoelectric half-space is discussed. Starting from the known form of the Green's function expressed in the spectral domain, the singular contributions are isolated and treated separately. It is found that the surface acoustic wave contributions (i.e., poles in the spectral Green's function) give rise to an anisotropic generalization of the Hankel function H0(2), the spatial Green's function for the scalar two-dimensional wave equation. The asymptotic behavior at infinity and at the origin (for the electrostatic contribution) also are explicitly treated. The remaining nonsingular part of the spectral Green's function is obtained numerically by a combination of fast Fourier transform and quadrature. Illustrations are given in the case of a substrate of Y-cut lithium niobate.