Very interesting... thanks a lot for the pointer JacquesC.

I will digest the readings there on MGfun...

I just wanted to clarify a bit: I am not taking the Fourier Transform of log(u)/u, instead, we are taking the Fourier transform of a piecewise function:

f1(u)= log(u)/ u, when u>1.0;

       = 0, when 0<=u<=1.0.

f2(u)= log(-u)/u, when u<-1.0

       =0, when 0>=u>=-1.0.

f3(u)= f(u) - f1(u) - f2(u),

I plan to take closed-form (yet involving extended functions and weird functions such as dirac delta, heaviside etc) Fourier Transform of f1 and f2, and use FFT to take Fourier Transform numerically for f3.

Then I will add them up together.

Why do I want to take its Fourier Transform even though it's not well behaved?

Taking Dirac delta as an example, it's not a traditional function, but it actually gives very nice properties in subsequent algebraic calculations...

And FFT does not work for the orignal f(u), because it is only square integrable, but not absolutely integrable.

FFT only works for traditional Fourier transform, with no weird functions, etc.

That's why we have to separately treat these three parts...

Of course, overall, our ultimate goal is to take FT of f(u), or numerically, FFT of f(u).

But if f(u) is not absolutely integrable, FFT doesn't work at all...

That's why we have to do the detours...

Any more thoughts?

Thanks a lot!