For a normal $ \frac 1 { x ^ 2 } $ function, the Fourier Transform is simply the delta Function, with a constant in front. It took me a while to understand the reason for this and the power of derivatives.
I can ask the question two ways:
- Is it possible to split the Fourier Transform in half, the negative side and the positive side and how would I do this?
- Let us make the inverse square function asymmetric, the following way, where $ a \ne b $: $$ f ( x ) = \begin {cases} \frac 1 { a x ^ 2 } & x < 0 \\ \frac 1 { b x ^ 2 } & x > 0 \end {cases} $$ I would like to take the Fourier Transform of this second function.
For me right now, this is an intellectual exercise. For you background, I have degrees in Physics, Mathematics, and Computer Science. It has been a LONG time since I had to do a Fourier Transform. After reading many web sites, many times, I have an understanding of the Fourier Transform of $ \frac 1 { x ^ 2 } $; but not if it is asymmetric.