In this work, the simplest partial differential equation in { bb R}^N is studied and some of its properties derived. It describes the infinitesimal translation of a function Φ in a fixed direction specified by a unit vector n and driven by a source function ρ. We work out several forms of the corresponding Green s function and show that the solutions may be viewed as integral transforms of ρ, known as divergent beam x-ray transform in imaging science. Physically, this connection is simply due to straight line propagation of radiation emitted by the spatial source distribution ρ. In particular, we examine the special two-dimensional case to point out the connection with the classical Radon transform. We then show how the Radon transform inversion can be obtained in the context of a complex extension of this equation. Perspectives in higher dimensional space, based on the present approach, are given in the conclusion.

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