Could you please explain why Mathematica gives the following expression when taking Fourier transform of $\exp(\lambda z)/\lambda$? $$\frac{-\log(-z)+\log(z)}{\sqrt{2\pi}}$$ Why the answer does not depend on $x$?

The source code is as the following:

In[1]:= FourierTransform[Exp[λ z]/λ,λ,x]
Out[1]= (-Log[-z]+Log[z])/Sqrt[2 π]
  • 2
    $\begingroup$ I'll venture a guess. The FT does not exist, the FT code fails to catch that, calls Integrate with GenerateConditions->False, and gets the strange result in that way. $\endgroup$ May 28 '16 at 15:32
  • 1
    $\begingroup$ @DanielLichtblau That can't be right; if FourierTransform calls Integrate, there wouldn't be any point of having FourierTransform in the first place. $\endgroup$
    – QuantumDot
    May 28 '16 at 17:49
  • $\begingroup$ @QuantumDot that can't be right: FourierTransform is very complex, and it may call Integrate in some specific cases... $\endgroup$ May 28 '16 at 18:39
  • $\begingroup$ @AccidentalFourierTransform Integrate is a more general function than FourierTransform. If anything, I would expect Integrate to make calls to FourierTransform for classes of integrals that are of the Fourier form -- not the other way around. $\endgroup$
    – QuantumDot
    May 28 '16 at 18:52
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    $\begingroup$ @AccidentalFourierTransform and Jason-with-the-even-longer-moniker are correct. FT sometimes calls Integrate, never the reverse. Integrate itself should probably use more table lookup, like the FT code does, but that's a separate issue. $\endgroup$ May 28 '16 at 21:36

Some commenters mentioned that the strange expression that Mathematica gives as a Fourier transform of Exp[\[Lambda] z]/\[Lambda] is a bug. If you are curious what the actual Fourier Transform is, you may find some insights in the related questions (e.g. link).

  • $\begingroup$ That FT does not exist. And that link involves unrelated FTs. $\endgroup$ Jul 4 '16 at 19:28

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