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Can you please help with the following. I would like to compute the sign of an expression:

Sign[Sqrt[2] σ + 
  E^((p - v0)^2/(2 σ^2))
    Sqrt[π] (p - v0) Erfc[(-p + v0)/(Sqrt[2] σ)]]

All 3 variables p, v0 and sigma are positive. Is there a way in which I can actually evaluate this expression?

I have deduced that for p higher or equal to v0 this expression is positive.

How can I check the sign of this expression when p is smaller than v0? For instance, assuming p goes to zero and sigma is 5 times smaller than v0?

Thanks a lot for your support, highly appreciated.

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As $\sigma>0$ we can divide the expression by $\sigma$ without changing its sign. Then, defining $x=(p-v_0)/\sigma$, the expression becomes

Sign[Sqrt[2] + E^(x^2/2)*Sqrt[π]*x*Erfc[-x/Sqrt[2]]]

This expression seems to be positive for any $x\in\mathbb{R}$, so I'd say that the answer to your question is that your Sign[...] is always 1:

LogPlot[Sqrt[2] + E^(x^2/2)*Sqrt[π]*x*Erfc[-x/Sqrt[2]], {x, -1000, 10},
  WorkingPrecision -> 100, PlotRange -> All]

enter image description here

The asymptotes of your expression are

  • $\sqrt{2}/x^2$ for $x\to-\infty$, which is positive,
  • $2x\sqrt{\pi} e^{\frac{x^2}{2}}$ for $x\to+\infty$, which is positive.
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  • $\begingroup$ @Roman, thank you so much. I managed to compute a limit by assuming p=0, sigma>0, as v0 goes to infinity and I get -sqrt(2)*sigma. Your solution is more elegant, thanks! $\endgroup$
    – Astrid
    Jan 17 at 13:18
  • $\begingroup$ @Daniel, thank you. I think it should either be multiplied by (-x) or have -x inside the Erfc function. Otherwise it returns -infinity. $\endgroup$
    – Astrid
    Jan 17 at 13:23
  • $\begingroup$ Sorry, I had made a typo, it should read : Erfc[ - x/(Sqrt[2])] and then: Limit[Sqrt[2] + E^(x^2/2) Sqrt[\[Pi]] x Erfc[-x/(Sqrt[2])], x -> -Infinity]what gives zero. I will delete the wrong comment. $\endgroup$ Jan 17 at 13:48
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It seems that the sign of the expression is more complex, it is not always positive.

Clear[f, x0];
f[x_] := Sqrt[2] + E^(x^2/2)*Sqrt[π]*x*Erfc[-x/Sqrt[2]];
NMinimize[f[x], x];
x0 = %[[2, 1, 2]]
f[x0]
Plot[f[x], {x, x0 - 1, x0 + 1}, PlotPoints -> 400]
FindRoot[f[x], {x, x0}]

{-1.46626, {x -> -38.5039}}

{x -> -38.3605}

enter image description here

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  • $\begingroup$ No, what you're seeing is an artefact of using insufficient WorkingPrecision. $\endgroup$
    – Roman
    Jan 18 at 8:30

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