Question: How to take Fourier Transform of log(t)/t ?

Hi,

I have difficulty in taking fourier transform of a function f(t).

The function f(t) is not integrable. Through analytical analysis, f(t)
behaves asymptotically the same as a*log(t)/t

for t-> +infinity; and f(t) behaves asymptotically the same as b*log(-
t)/t

for t-> -infinity; where "a" and "b" are some constants.

Is there a way to work around this difficulty and get some sort of
fourier transform of this function f(t), possibly

involving extended functions or functions which can only be evaluated
numerically or distribution functions...

Any thoughts?

Thanks a lot!

 It seems Maple cannot do it... I don't know why...

------------------

It looks like Mathematica can do it, but gave very hyper results:

\text{FourierTransform}[\text{Log}[x]/x, x, w]

-\frac{1}{2} \sqrt{\frac{\pi }{2}} (2 i \text{EulerGamma}+\pi +2 i
\text{Log}[\text{Abs}[w]]) (-1+\text{Sign}[w])

\text{FourierTransform}[\text{Piecewise}[\{\{0, x\text{$<$=}1\}, \
{\text{Log}[x]/x, x>1\}\}], x, w]

\frac{1}{216 \sqrt{2 \pi } \text{Abs}[w]}\left(-108 i \pi  w
(\text{EulerGamma}+\text{Log}[\text{Abs}[w]])+\text{Abs}[w] \left(24 i
w^3 \text{HypergeometricPFQ}\left[\left\{\frac{3}{2}\right\},\left\
{\frac{5}{2},\frac{5}{2}\right\},-\frac{w^2}{4}\right]+8 i w^3
\text{HypergeometricPFQ}\left[\left\{\frac{3}{2},\frac{3}{2}\right\},
\left\{\frac{5}{2},\frac{5}{2},\frac{5}{2}\right\},-\frac{w^2}
{4}\right]+9 \left(12 \text{EulerGamma}^2-\pi ^2-3 w^2
\text{HypergeometricPFQ}\left[\{1,1,1\},\left\{\frac{3}{2},2,2,2\right
\},-\frac{w^2}{4}\right]+\text{EulerGamma} \text{Log}
\left[w^{24}\right]+12 \text{Log}[\text{Abs}[w]]^2+24 i \text{Sin}[w]
\right)\right)\right)

Any thoughts? Thanks!

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My idea is that if we can find the closed form transform of log(u)/u , possibly involving extended functions or "weird" functions,

such as Dirac delta function or Heaviside function, etc.

Then we can separate the original function f(t) into two parts, one is absolutely integrable part, on which we can do FFT , and the other part is the square integrable part, on which we hopefully obtain the closed form expression...

Is this the most appropriate approach?

Thanks