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I want to do a Fourier transform to the below function by Mathematica. How can I do it? Here $c$, $d$, $a$, $L$ are constants.

$$ w(r)= \left\{ \begin{array}{ll} -\frac{c}{\epsilon r} & r> L \\ -\frac{c}{\epsilon d}\left[\frac{a}{r}-\ln\left(\frac{r}{L}\right)\right] & a < r\leq L \\ -\frac{c}{\epsilon d}\left[\ln\left(\frac{a}{L}\right)\right] & r \leq a \\ \end{array} \right. $$

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It looks like Mathematica can do this. Define something like this (I've simplified your version slightly)

w[r_] := Piecewise[{{-1/r, r > L}, {-1/d (a/r - Log[r/L]), 
    a < r <= L}, {-1/d Log[a/L], r <= a}}]

Mathematica will return a continuous Fourier transform

Assuming[0 < a < L, FourierTransform[w[r], r, k]]

I suggest you check the help for FourierTransform to ensure that the definition used is the one you want.

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  • $\begingroup$ Thank you @mikado! Is this 'Assuming[0 < a < L, FourierTransform[w[r], r, k]]' code only for the travel from 0 to L, exclude the r>L and r<= L? $\endgroup$
    – aluuzz
    Commented Sep 8, 2019 at 22:50
  • $\begingroup$ No, I'm just saying that a is positive and less than L $\endgroup$
    – mikado
    Commented Sep 9, 2019 at 5:55

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