= Abel inversion of stellar acoustic travel times
{c}
{title2=$\ln(R/r)=\frac{2a}{\pi}\int_0^a\frac{wF'(w)}{\sqrt{a^2-w^2}}dw$}
Assume $a=c/r$ decreases outward, tends to zero at $R$, and define $h(a)=-d\ln r/da$. Differentiation of the <Duvall law> gives $F'(w)=\int_0^w h(b)b/[w^2\sqrt{w^2-b^2}]\,db$. Interchanging <integrals> and using $\int_b^a dw/[w\sqrt{w^2-b^2}\sqrt{a^2-w^2}]=\pi/(2ab)$ proves the displayed <Abel transform> inversion. For a finite outer reference value $w_s$, both lower limits become $w_s$, after removing the known outer-layer contribution to the travel time. Inversion requires a monotone branch and amplifies errors in differentiated frequency data.
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