 Solution

Define the ingoing coordinateThen $dt=dv-dr/f$ andThis is the higher-dimensional analogue of Ingoing Eddington-Finkelstein coordinates. Its metric and inverse are regular at $r=r_{+}$, including when the zero is degenerate. A gauge transformation removes the singular radial term in the transformed potential, leaving the regular representative $A=-Q{r}^{-(d-3)}dv$. These expressions analytically extend the spacetime through the future outer horizon.