Computing the sign of an expression

Question

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.

Answer 1

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]

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.

Answer 2

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}