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Is there a documentation for the function Simplify`PWToUnitStep? This is a function which can be used to express a Piecewise function as a sum of UnitStep functions. I used it in the following way (in the last line of the provided code):

hfun[h_, J1_, J2_] := Sign[h] (Abs[h] - (Abs[J2] - Abs[J1]))/2
Jfun[h_, J1_, J2_] := (Abs[J2] + Abs[J1] - Abs[h])/2
Field1Evolution[h_, J1_, J2_] := 
 Piecewise[{{Sign[h] J1, 
    Abs[J2] >= Abs[J1] && 
     Abs[h] >= Abs[J1] + Abs[J2]}, {Sign[J1] hfun[h, J1, J2], 
    Abs[J2] >= Abs[J1] && (Abs[J2] + Abs[J1]) > 
      Abs[h] && (Abs[J2] - Abs[J1]) < Abs[h]}, {0, 
    Abs[J2] >= Abs[J1] && (Abs[J2] - Abs[J1]) >= Abs[h]}, {Sign[h] J1,
     Abs[J1] > Abs[J2] && 
     Abs[h] >= Abs[J2] + Abs[J1]}, {Sign[J1] hfun[h, J1, J2], 
    Abs[J1] > Abs[J2] && (Abs[J2] + Abs[J1]) > 
      Abs[h] && (Abs[J1] - Abs[J2]) < Abs[h]}, {Sign[J1] h, 
    Abs[J1] > Abs[J2] && (Abs[J1] - Abs[J2]) >= Abs[h]}}]

Simplify`PWToUnitStep[Field1Evolution[h, J1, J2]]

but the output includes some terms like 1 - UnitStep[-Abs[J1] + Abs[J2]] instead of simply UnitStep[+Abs[J1] - Abs[J2]], I know that the two expressions are the same but I would like the second as output, for some reasons. Can someone help me in getting only terms with UnitStep instead of `1-Unitstep`?

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    $\begingroup$ Your premise is wrong. Take a look at the documentation of UnitStep: It is equal to $0$ for $x<0$ and $1$ for $x\geq0$. Hence 1 - UnitStep[x] is not equivalent to UnitStep[-x], because of the discontinuity at $x=0$: 1 - UnitStep[0] == 0 while UnitStep[-0] == 1. You can also take a look at this image to see the transformations that the undocumented (!) function Simplify`PWToUnitStep makes. $\endgroup$ Commented Apr 30, 2024 at 16:42
  • $\begingroup$ Oh thanks @Domen. You basically answered my question. For my purposes >= or > is the same, thus puttin >= or <= every time I had > or <, respectively, I would get UnitStep instead of 1-UnitStep. I am sorry for my error, thank again! $\endgroup$ Commented Apr 30, 2024 at 16:54
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    $\begingroup$ And Simplify`PWToUnitStep is undocumented, we have posts listing useful undocumented functions though: mathematica.stackexchange.com/a/133530/1871 $\endgroup$ Commented May 1, 2024 at 4:28
  • $\begingroup$ At least we may have a look at kitchen by ?"Simplify`*". Click on any of the entries in that long list, there is no documentation. $\endgroup$ Commented May 1, 2024 at 5:06
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    $\begingroup$ @RuthMurphy, note that you likely won't be able to completely avoid 1 - UnitStep, beacuse sooner or later you need the negation of $\geq$ or $\leq$, which leads to $<$ and $>$. Take a look at this example: s = Simplify`PWToUnitStep[Piecewise[{{1, a >= b}, {2, b >= c}}]]. The first term in the output represents $a \geq b$, but the second term correctly represents $a<b \wedge b\geq c$. If you blindly replace 1 - UnitStep[x] with UnitStep[-x], you will get wrong result for the edge cases: {s, s /. 1 - UnitStep[x_] :> UnitStep[-x]} /. {a -> 1, b -> 1, c -> 1} results in {1, 3}. $\endgroup$ Commented May 1, 2024 at 7:40

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