= Critical-point Jensen identity for the hyperbolic derivative
{title2=$T_{\mathbb D}(R)+N_{\rm crit}(R)=\langle\log H_g\rangle_R-\log H_g(0)$}
For $g'(0)\ne0$, apply <Jensen's formula> with the smooth curvature term to the <hyperbolic derivative density>. Its distributional <Laplacian> is $H_g^2+2\pi\sum m_a\delta_a$, giving the displayed identity with <ramification points> defined by $g'(a)=0$. If $g'(z)=cz^m+\cdots$ at zero, use $m\log R$ as the origin counting term and replace the base constant by $\log\lim_{z\to0}H_g(z)/|z|^m$. Without this regularization the basepoint-critical formula is undefined.
Back to article page
