simplifying expressions - Assumptions on unknown functions

I'm trying to do analytic calculations in a quantum mechanic harmonic oscillator basis. Specifically I want to be able to evaluate functions of the many particle density.

I define the following functions: [definitions]

(* Harmonic Oscillator Eigenstate *)
basis[i_, z_] = Pi^(-1/4)/Sqrt[2^i Factorial[i]] Exp[-z^2/2] HermiteH[i, z];
(* Orbital *)

orb[i_, z_] = Sum[c[i, k] basis[k, z], {k, 0, t}];
(* Particle density of one orbital *)
orbitaldens[i_, z_] = Conjugate[orb[i, z]] psi[i, z];
(* Total density *)
dens[z_] = Sum[a[i] orbitaldens[i, z], {i, 1, n}];

To help Mathematica I define some assumptions beforehand: [assumptions]

$Assumptions = Element[z, Reals] && (* Real-space coordinate is real *)
               Element[{n, t}, Integers] && n > 0 && t > 0 && (* Just indices *)
               Element[c[i_, j_], Complexes] && (* Complex state vector *)

               Element[a[i_], Reals]; (* Real filling factor *)

What is the correct way to define assumptions on an unknown function? Here c[i_, j_] should take integers i, and j and return a complex number. a[i_] on the other hand should take on integer i and return a positive real number.

Unfortunately, the evaluation of the definitons takes very long if I include these assumptions. Without them the definitions block takes about a second to evaluate.

However, I need those assumptions, so Mathematica can do useful simplifications. For example Conjugate[orb[i,z]] orb[i,z]//FullSimplify should simplify to Abs[orb[i,z]]^2. But the same should work, if I only apply the complex conjugation to c[i,k] inside orb[i,z], because basis[i,z] is real. So far, this takes too long to evaluate.

Is it possible to speed this whole thing up?

Another problem I found is the codomain of the Factorial The following two expressions do not evaluate to True, neither to False. They just repeat the first parameter of Refine:

Refine[Element[Sqrt[Factorial[i]], Reals], Element[i, Integers] && i > 0]
Refine[Factorial[i] > 0, Element[i, Integers] && i > 0]

In words, the factorial of a positive integer is not necessarily positive.

What is wrong there? I don't see how the factorial of a positive integer could become negative. (In symbolic maths)

Comments

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Suppose we have the following simple RegionPlot: f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}] Now I'm trying to change the curve defined by $y=g[x]$ into a thick black curve, while leaving all other boundaries in the plot unchanged. I've tried adding the region $y=g[x]$ and playing with the plotstyle, which didn't work, and I've tried BoundaryStyle, which changed all the boundaries in the plot. Now I'm kinda out of ideas... Any help would be appreciated! Answer With f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 You can use Epilog to add the thick line: RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}, PlotPoints -> 50, Epilog -> (Plot[g[x], {x, 0, 2}, PlotStyle -> {Black, Thick}][[1]]), PlotStyle -> {Directive[Yellow, Opacity[0.4]], Directive[Pink, Opacity[0.4]],

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