 Solution

For $z>0$, $\varphi =q(4\pi {\epsilon }_0{)}^{-1}(|x-d\hat{z}{|}^{-1}-|x+d\hat{z}{|}^{-1})$ and $E=-\nabla \varphi$. The normal field just above the plane is $-qd/[2\pi {\epsilon }_0(r^2+d^2{)}^{3/2}]$, so $\sigma ={\epsilon }_0{E}_z$ is the stated expression. Integrating $2\pi r\sigma (r)dr$ from zero to infinity gives total induced charge $-q$.