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John's equation is an ultrahyperbolic partial differential equation satisfied by the X-ray transform of a function. It is named after Fritz John.
Given a function f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} } with compact support the X-ray transform is the integral over all lines in R n {\displaystyle \mathbb {R} ^{n}}. We will parameterise the lines by pairs of points x , y ∈ R n {\displaystyle x,y\in \mathbb {R} ^{n}} , x ≠ y {\displaystyle x\neq y} on each line and define u {\displaystyle u} as the ray transform where
Such functions u {\displaystyle u} are characterized by John's equations
which is proved by Fritz John for dimension three and by Kurusa for higher dimensions.