 Conformal inversion of a quadrant

The map $w=-1/z$ sends the open second quadrant inside the unit circle bijectively to the open first quadrant outside it. Radius becomes $r^{-1}$ and argument becomes $\pi -\theta$. Its derivative $1/z^2$ is nonzero on the source, so it is a conformal map. Rotation combined with reciprocal inversion similarly maps other quarter-disc regions.