It is shown, on the basis of potential theory, that for a Regge pole with position α(t)>12 the reduced residue β(t)να(t) behaves essentially as [α'(t)R]R2α(t) near threshold (ν<~0), where R is the effective radius interaction. The quantity γ(t)=(m02ν)αβ(t) which occurs in the asymptotic term γ(t)(s2m02)α can therefore be considered as a slowly varying function of t only if one takes m0=R-1. Assuming that our threshold expression gives a qualitative description in the relativistic case, we note that if Mr>1, where M is the nucleon mass, then the normalization m0=M used in high-energy phenomenology will give rise to an exponential falloff of γ(t) with (width)-1~2α'(0) ln (MR). The values of R in the t channel of ππ, πN, and NN, estimated from the knowledge of the nearest left-hand branch points or factorization, occur in increasing order, and for each of them MR>1. Our results for the Pomeranchuk trajectory roughly reproduce the exponential diffraction shape and indicate that a larger total cross section implies a sharper falloff of the amplitude in the diffraction region. A connection between f0 and the diffraction width is discussed, as well as the question of zeros in βνα.