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I was wondering if it is easy to see if a given expression evaluates to a real number in a given domain? Let $f(x,y,z)$ be some function depending on three variables $x,y,z$. I would like to know if $f(x,y,z)$ is real in some domain, e.g., $x > 0, y < 0, \vert z \vert < 1$ or $x > 0, y < 0, z < g(x,y)$ (the last condition on $z$ here just means for a given $x$ and $y$, $z$ must be smaller than the value $g(x,y)$).

I've tried a few things already but it seems I can't get it to work without specifying my variables from the outset (in which case I would then see if the function was real by eye). However, for many different $(x,y,z)$-tuples this becomes tedious and I would like to see if it's possible to see if a function is real for all points in some domain within Mathematica.

Let

f(x,y,z) = PolyLog[2, x*y/z] - PolyLog[2, -x*y/z]

be a typical function I am dealing with.

I tried

Reduce[PolyLog[2, x*y/z] - PolyLog[2, -x*y/z] ∈ Reals && x > 0 && y < 0 && z < 1]

but this system cannot be solved by Reduce. I've also tried

FunctionDomain[PolyLog[2, x*y/z] - PolyLog[2, -x*y/z], {x, y, z}, Reals]

but it did not help either. I am using version 11.1.1.

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    $\begingroup$ You can try FunctionDomain and see if it works for you. $\endgroup$ – Nasser Jul 11 at 11:51
  • $\begingroup$ @Nasser Thank you. I have tried this and I get the error 'Unable to find the domain with the available methods'. It seems this function FunctionDomain cannot cope with the PolyLog[2,x], which is part of my f(x,y,z). $\endgroup$ – CAF Jul 11 at 12:57
  • $\begingroup$ With version 12.1.1 FunctionDomain[PolyLog[2, x], x] evaluates to x <= 1 $\endgroup$ – Bob Hanlon Jul 11 at 13:38
  • $\begingroup$ @BobHanlon Thanks, I see, I am unfortunately using version 11.1.1 $\endgroup$ – CAF Jul 11 at 15:13
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    $\begingroup$ First, FunctionDomain[PolyLog[2, x], x, Reals] performs x<1 in version 12.0. Second, this is an empty talk without knowing your f(x,y,z). $\endgroup$ – user64494 Jul 11 at 17:25
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Just get an overview with RegionPlot3D .

RegionPlot3D[
    PolyLog[2, x*y/z] - PolyLog[2, -x*y/z] \[Element] Reals && x > 0 && 
      y < 0 && z < 1, {x, -3, 3}, {y, -3, 3}, {z, -3, 3}, 
PlotPoints -> 100]

enter image description here

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