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The integral of a source over a directed line, weighted by exponential absorption between each source point and a detector. For line coordinates $x=p\mathbf{n}+s\mathbf{d}$ and a detector at the negative end, it is $\int f(p,s)\exp [-{\int }_{-\infty }^s\mu (p,r)dr]ds$. The integrating factor of a transport equation derives the weight. Reversing the detector direction changes the weight for spatially varying attenuation, so the two directed measurements need not coincide. Setting $\mu =0$ recovers the Radon transform.

For $x_{n+1}=a_n{x}_n+b_n$, put $U_0=1$ and $U_n={\prod }_{j=1}^n{a}_j$, with $a_j\ne 0$. Dividing by this accumulated homogeneous growth gives $z_n=x_n/U_{n-1}$ and $z_{n+1}-z_n=b_n/U_n$. A telescoping series then provesThis is the discrete counterpart of the integrating factor for a first-order linear differential equation.