Radon Transform--Cylinder

Let the two-dimensional cylinder function be defined by

Then the Radon transform is given by

where

is the delta function. Rewriting in polar coordinates then gives

Now use the harmonic addition theorem to write

with a phase shift. Then

Then use

which, with , becomes

Define

so the inner integral is

and the Radon transform becomes

Converting to using ,

which could have been derived more simply by