Prandtl–Meyer function

Remarks

Consider a plane, steady, isentropic supersonic flow turned through a small angle $d\theta$ across a Mach wave. In an expansion its Mach number changes so that $d\nu = d\theta$, and this is where the function comes from. Example: for $\gamma = 1.4$ and $M = 2$, one has $\sqrt{6}\arctan\sqrt{1/2} - \arctan\sqrt{3} \approx 2.449 \cdot 35.26^\circ - 60^\circ \approx 26.38^\circ$. The function is strictly increasing, and $\nu(M) \to \nu_{\max} = \frac{\pi}{2}\left(\sqrt{\frac{\gamma + 1}{\gamma - 1}} - 1\right)$ as $M \to \infty$, which is about $130.45^\circ$ for $\gamma = 1.4$. Non-example: for $M < 1$ the expression is not defined over the reals, since $\sqrt{M^2 - 1}$ is imaginary; subsonic flows do not turn by simple waves.