# Fourier transform of Exp[x]/x

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:= FourierTransform[Exp[λ z]/λ,λ,x]
Out= (-Log[-z]+Log[z])/Sqrt[2 π]

• 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. – Daniel Lichtblau May 28 '16 at 15:32
• @DanielLichtblau That can't be right; if FourierTransform calls Integrate, there wouldn't be any point of having FourierTransform in the first place. – QuantumDot May 28 '16 at 17:49
• @QuantumDot that can't be right: FourierTransform is very complex, and it may call Integrate in some specific cases... – AccidentalFourierTransform May 28 '16 at 18:39
• @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. – QuantumDot May 28 '16 at 18:52
• @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. – Daniel Lichtblau 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).