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

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Discussion Overview

The discussion centers on the challenge of solving the inverse Fourier transform for the function 1/w^2, particularly in the context of functions that can be Fourier transformed despite not being absolutely integrable. Participants explore methods for addressing the high order pole at the origin and the implications for the inverse transform.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant notes that while a function like |t| can yield a Fourier transform of 1/w^2, the inverse transform presents difficulties due to the high order pole at the origin.
  • Another participant suggests integrating around the pole or using knowledge of the forward transform as potential strategies for solving the inverse transform.
  • A different participant questions the applicability of the Cauchy principal value, stating it is typically valid only for first order poles.
  • Some participants mention that high order poles can sometimes be managed, but if this specific case cannot be resolved, leveraging the known forward process remains an option.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of handling high order poles in the context of the inverse Fourier transform, indicating that multiple competing approaches exist without a clear consensus on the best method.

Contextual Notes

The discussion highlights the limitations of applying the Cauchy principal value to higher order poles and the unresolved nature of the integral involved in the inverse transform.

jtceleron
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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.
 
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Integrate around the pole - or exploit the fact you already know the forward transform.
 
Simon Bridge said:
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.
 
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.
 

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