= Herglotz–Wiechert inversion
{c}
{title2=$r(\mu)=a\exp[-\pi^{-1}\int_\mu^{\eta_a}\Delta(p)(p^2-\mu^2)^{-1/2}\,dp]$}
For a spherically symmetric speed with strictly increasing $\eta(r)=r/v(r)$, a turning ray of parameter $p$ has $\Delta(p)=2p\int_p^{\eta_a}(d\log r/d\eta)(\eta^2-p^2)^{-1/2}\,d\eta$. Integrate this against $(p^2-\mu^2)^{-1/2}$ and interchange the triangular integration region. The inner kernel integral is $\pi/2$, giving the displayed radius and then $v(r(\mu))=r(\mu)/\mu$. Complete turning-ray branch data are required.
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