Mathematica gives the derivative of the function -I (Log[-x] - Log[x]) as $0$, but on the real domain the expected result is $\pi\delta(x)$ and on complex domain it is much more complicated. Mostly on the complex plane the function is smooth, differentiable and non-constant. Why do I get its derivative zero? How can I get the true expression for the derivative?

The function can be represented as

$$-i(\ln(-x)-\ln x)=2 \arg(-x)-\pi$$

This is the plot of the desired derivative: enter image description here

It was obtained with the following command:

ComplexPlot3D[-2 Derivative[1][Arg][-x], {x, -3 - 3 I, 3 + 3 I}, WorkingPrecision -> 100]

It is zero only along the real axis.

  • $\begingroup$ @MariuszIwaniuk the derivative function cannot differentiate Sign[x]. Additionally, it is undefined for complex values. The original function is well defined on the whole complex plane. $\endgroup$ – Anixx Apr 2 at 10:33
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    $\begingroup$ That derivative is zero because of the chain rule. Which means it vanishes on C, not just R. $\endgroup$ – Daniel Lichtblau Apr 2 at 13:26

Mathematica doesn't yield results using generalized functions unless Fourier/Laplace transforms are involved. So, differentiate using a transform.

LaplaceTransform[-I (Log[-x] - Log[x]), x, s]
(* -I ((EulerGamma + Log[s])/s - (EulerGamma - I \[Pi] + Log[s])/s) *)
InverseLaplaceTransform[s %, s, x]
(* \[Pi] DiracDelta[x] *)
  • $\begingroup$ Okay. But what about the complex case? Dirac Delta is not defined on complex arguments. But the original function is differentiable. $\endgroup$ – Anixx Apr 2 at 11:41
  • $\begingroup$ @Anixx That is the complex case. $\endgroup$ – John Doty Apr 2 at 11:43
  • $\begingroup$ @Anixx What makes you think DiracDelta is undefined for complex arguments? $\endgroup$ – John Doty Apr 2 at 11:45
  • $\begingroup$ Try evaluate it in Mathematica $\endgroup$ – Anixx Apr 2 at 11:45
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    $\begingroup$ @Anixx Perhaps that behavior is a bug. You might want to look at this: mathoverflow.net/questions/118101/… $\endgroup$ – John Doty Apr 2 at 12:01

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