How to Solve the Inverse Fourier Transform for 1/w^2?

[jtceleron]

A necessary condition that a function f(x) can be Fourier transformed is that f(x) is absolutely integrable. However, some function, such as |t|, still can be Fourier transformed and the result is 1/w^2, apart from some coefficients. This can be worked out, as we can add a exponential attenuation factor, and then send it to 0. In physics, we are always doing such things.

However, the inverse transform is not so apparent, the how to solve the inverse Fourier transform for 1/w^2? Indirectly, we have already know the result. but directly, how to solve this integral? Because we have a high order pole at the origin. It seems the divergence cannot be avoided.

I am confused with that.

 

[Simon Bridge]

Integrate around the pole - or exploit the fact you already know the forward transform.

 

[jtceleron]

Integrate around the pole - or exploit the fact you already know the forward transform.

but I think the Cauchy principal value is available only when the pole is of first order.

 

[Simon Bridge]

Some high order poles can be dealt with though.
If this one cannot be, then you still have the ability to use the fact that you know the reverse process.