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For an absolutely continuous $f$ with $f(0)=0$, this causal Abel integral equation has inversionUnder Laplace-transform growth hypotheses, the convolution theorem for Laplace transforms converts the original kernel into $\Gamma (1-\alpha )s^{\alpha -1}$. Since $\mathcal{L}[f^{\prime }]=s\mathcal{L}[f]$, the transformed inverse is $s^{-\alpha }\mathcal{L}[f^{\prime }]/\Gamma (1-\alpha )$, giving the formula. It also holds locally in the corresponding integrable convolution class: convolving the two power kernels produces the constant $B(\alpha ,1-\alpha )=\Gamma (\alpha )\Gamma (1-\alpha )$ by the beta-gamma identity, so substitution recovers ${\int }_0^t{f}^{\prime }=f(t)$. The same convolution identity proves uniqueness in that class. For example $f(t)=t$ gives $\Phi =t^{\alpha }/[\Gamma (\alpha +1)\Gamma (1-\alpha )]$.

For the isotropic Jeans-Abel mass inversion, let $J(R)=I(R){\sigma }_{\mathrm{los}}^2(R)$ and $p_{\ast }(r)=\nu (r){\sigma }_r^2(r)$. The two preceding projections have the same Abel integral equation kernel, givingTheir ratio recovers ${\sigma }_r^2=p_{\ast }/\nu$, and the isotropic Jeans equation givesEquivalently put $A(r)={\int }_r^{\infty }{I}^{\prime }(R)/\sqrt{R^2-r^2}dR$ and $B(r)={\int }_r^{\infty }{J}^{\prime }(R)/\sqrt{R^2-r^2}dR$. Then $M=-r^2{B}^{\prime }(r)/(GA(r))$: the common stellar luminosity cancels from the inferred mass. Differentiating these singular-kernel integrals is understood through suitable regularity or a nonsingular variable change, not by naively keeping an infinite lower-end boundary term.

The inversion assumes adequate radial coverage, convergence and a smooth spherical equilibrium. In practice derivatives amplify noise and the unobserved outer tail needs treatment or a model. Unknown anisotropy reintroduces a mass–anisotropy degeneracy; the uniqueness just derived is conditional on the explicitly isotropic velocity distribution.

Let the surface radius be $a$, let $\Delta$ denote epicentral angle in radians, and put $\eta (r)=r/v(r)$. The spherical elastic ray invariant gives $\sin i=p/\eta$. Along either radial leg, if $l$ is arclength and $\theta$ polar angle in the ray plane,For a turning ray with minimum radius $r_p$ defined by $\eta (r_p)=p$, the outgoing and incoming legs therefore giveThe square-root turning singularity is integrable for a regular turning point.

Introduce the reduced travel timeIts integrand vanishes at the moving lower endpoint, so differentiation gives ${\tau }^{\prime }(p)=-\Delta (p)$. Consequentlyon every regular travel-time branch. This is also the endpoint variation of travel time: the tangential endpoint slowness multiplied by the surface radius is $p$. Angles expressed in degrees would introduce the corresponding conversion factor; the displayed equality uses radians.

For Herglotz–Wiechert inversion, assume $\eta$ is strictly increasing so that $r=r(\eta )$ is single-valued, and set ${\eta }_a=a/v(a)$ and $h(\eta )=d\log r/d\eta$. The angular integral becomes an Abel integral equation:For any $0<\mu <{\eta }_a$, multiply by $(p^2-{\mu }^2{)}^{-1/2}$ and integrate. Interchanging the triangular region $\mu <p<\eta <{\eta }_a$ givesThe inner integral equals $\pi /2$: the substitution $p^2={\mu }^2+({\eta }^2-{\mu }^2){\sin }^2\theta$ reduces it to ${\int }_0^{\pi /2}d\theta$. Thus the full constructive inversion isThe surface radius and surface slowness fix the normalization. Varying $\mu$ reconstructs radius and speed parametrically; the centre is recovered by a limit when the data extend that far. The calculation assumes smooth regular turning rays and complete branch data, not merely the first arrival at each epicentral angle.

For a check, a constant speed $v_0$ gives ${\eta }_a=a/v_0$, $\Delta (p)=2\arccos (p/{\eta }_a)$ and $T(p)=2\sqrt{a^2-p^2{v}_0^2}/v_0$. These satisfy $dT/d\Delta =p$, and the inversion returns $r(\mu )=v_0\mu$.

If $\eta$ decreases outwards on an interval, then it increases as a surface ray travels inward through that interval. Such a ray cannot turn there, since turning requires the decreasing inward value of $\eta$ to reach $p$. Rays that penetrate this interval can traverse it and turn deeper, but the map $\eta \mapsto r$ now has several branches. The angular data constrain combinations of those branch contributions, rather than the single function $h(\eta )$ used above. This is a hidden turning-ray zone from nonmonotone spherical slowness. The stated single-valued inversion fails and does not uniquely locate the speed profile in that interval without additional information. Missing turning rays and possible multiple travel-time branches are the obstruction, not a failure of the algebraic Abel-kernel identity.