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In a previous paper, we computed expressions for the Detweiler-Whitingsingular field of point scalar, electromagnetic and gravitational chargesfollowing a geodesic of the Schwarzschild spacetime. We now extend this to thecase of equatorial orbits in Kerr spacetime, using coordinate and covariantapproaches to compute expansions of the singular field in scalar,electromagnetic and gravitational cases. As an application, we give thecalculation of previously unknown mode-sum regularization parameters. We alsopropose a new application of high-order approximations to the singular field,showing how they may be used to compute $m$-mode regularization parameters foruse in the $m$-mode effective source approach to self-force calculations.

High-order expansions of the Detweiler-Whiting singular field in Kerr spacetime