Why Does Inverse Fourier Transform of Sinc Function Require Contour Integration?

[bdforbes]

I can easily find the Fourier transform of rect(x) to be $2sinc(2\pi k)$ using particular conventions (irrelevant here). But when I attempt to inverse Fourier transform the sinc function, I find I have to resort to contour integration and Cauchy principal values.

This is troubling to me. It appears as if the usual definition of a Fourier transform is inadequate here, and could possibly lead to incorrect results in another context. Can anyone shed any light on this?

 

[bdforbes]

I have come to the conclusion that the Cauchy Principal Value is the appropriate type of improper integral to use. This is because Fourier transforms originate in the consideration of periodic functions; we should consider some approximation to the sinc function which is periodic, i.e. cut the function off at some finite symmetric boundary, take the Fourier transform, and then take the limits symmetrically to infinity.