Kuramoto self-consistency equation ($1=K{\int }_{-\pi /2}^{\pi /2}{\cos }^2\theta g(Kr\sin \theta )d\theta$)

For a symmetric stationary frequency distribution, locked oscillators contribute $r={\int }_{-Kr}^{Kr}\sqrt{1-{\omega }^2/(Kr{)}^2}g(\omega )d\omega$. The drifting oscillators' real contribution is zero by integrating a periodic logarithmic derivative, and imaginary contributions cancel by symmetry. Substituting $\omega =Kr\sin \theta$ and dividing by $r>0$ gives the equation. The incoherent solution must be retained before division. For finite positive continuous $g(0)$ the branch reaches it at $K_c=2/[\pi g(0)]$.