SUMMARY I present, in a rigorous form, the Green’s functions that represent the dynamic displacement response of an infinite, homogeneous and isotropic medium to a constant slip rate on a quadrantal fault that continues perpetually after the slip onset. I also present analytical expressions for the corresponding stress Green’s functions that are a major simplification over previously known solutions. The equations derived can be utilized in the formulation of boundary element methods that discretize a fault plane into a set of small rectangular elements and assume the slip rate to be piecewise constant within each discrete element. 1I N T R O DUCTION In the boundary element method (BEM) modelling of the elastodynamic field created by slip on a fault in a three-dimensional (3-D) medium, the fault plane is often discretized into a set of small rectangular elements, on each of which the slip is assumed to follow certain types of spatio-temporal variation profiles. One of the simplest and most popular categories of such assumption is that of the so-called piecewise constant slip rate, which states that the slip rate takes a constant value everywhere and at every moment within a given rectangular element and within a given discrete time window. The elastodynamic field in the medium is calculated by convolving the discrete spatio-temporal profile of the fault slip with Green’s functions corresponding to a constant slip-rate that takes place on each discrete fault element and within each discrete time window. The knowledge of these Green’s functions therefore makes an indispensable part of the BEM modelling of fault dynamics. In mathematical terms, a constant slip-rate profile localized on a rectangular fault element and within a finite time window is equivalent to a superposition, with appropriate offsets in space and time and with appropriate sign reversals, of eight identical profiles of slip rate that takes a constant value everywhere on a quadrantal part of the fault and at every moment after the onset of slip (Fukuyama & Madariaga 1998). The question of calculating Green’s functions for a constant slip rate on a rectangular space‐time element thus reduces to the question of calculating the Green’s functions for a constant slip rate on a quadrantal fault that goes on perpetually after the onset of slip. The stress Green’s functions for such a quadrantal fault model placed in an infinite, homogeneous and isotropic medium were obtained analytically by Aochi (1999) and Aochi et al. (2000a) and were fully utilized in their numerical simulation of fault dynamics. The analytical expressions for the displacement Green’s functions have remained unknown, however, because of the complexity of the algebra involved. In the present study I have derived, in a rigorous form, Green’s functions that represent the displacement response of an infinite, homogeneous and isotropic medium to a constant slip rate on a quadrantal fault that continues perpetually after the slip onset. Although these Green’s functions can be evaluated numerically, the use of analytical expressions is highly beneficial from the viewpoint of computational efficiency. I have double-checked my equations by deriving them with two different methods and confirming their consistency with a corresponding two-dimensional (2-D) theory. I have also been able to substantially simplify the expressions for the stress Green’s functions over those obtained by Aochi (1999) and Aochi et al. (2000a).