A note on Laplace's equation inside a cylinder

Two difficulties connected with the solution of Laplace's equation around an object inside an infinite circular cylinder are resolved. One difficulty is the non-convergence of Fourier transforms used, in earlier publications, to obtain the general solution, and the second difficulty concerns the existence of apparently different expressions for the solution. By using a Green's function problem as an easily analyzed model problem, we show that, in general, Fourier transforms along the cylinder axis exist only in the sense of generalized functions, but when interpreted as such, they lead to correct solutions. We demonstrate the equivalence of the corrected solution to a different general solution, also previously published, but we point out that the two solutions have different numerical properties.