= Projection formula for inverse attenuated Radon transform
{title2=$g=(4\pi)^{-1}(\partial_{x_1}-i\partial_{x_2})\int_0^{2\pi}e^{i\theta}J\,d\theta$}
Let $F(\rho,\tau,\theta)=\int_\tau^\infty f(s\boldsymbol d+\rho\boldsymbol n)ds$, $S=\mathcal Rf$, and $A=\int_{\mathbb R}e^{-F}g\,d\tau$ be the directed <attenuated Radon transform>. With the <signed Cauchy boundary operators>, set $J=e^F[e^{P^-S}P^-(e^{-P^-S}A)+e^{-P^+S}P^+(e^{P^+S}A)]$. An <integrating factor> turns the attenuated equation into unattenuated directional transport; its spectral solution has jump $-J$ on the unit circle. A normalized <Cauchy integral formula> and the coefficient at spectral infinity then prove the reconstruction formula. Known attenuation and full directed angular data determine the source; attenuation itself is not reconstructed by this pair.
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