The theory of spherical functions for the Lorentz group, required for harmonic analysis on the hyperboloid model of 3-dimensional hyperbolic space sitting in Minkowski space is considerably easier than the general theory. It only involves representations from the spherical principal series and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on R . This theory is discussed in Takahashi (1963), Helgason (1968), Helgason (2000) and the posthumous text of Jorgenson & Lang (2008).