White function or quadratic cosine "comes"

Some people on the forum told me that solutions to this problem cannot exist, since the square and round ones do not appear to each other (as I understood it), but after experimenting a little with plotting on WolframAlpha, I decided that this is a fundamentally wrong approach. As it turned out, the whole thing is just in "quantum entanglement." But first things first.



How to model entanglement? We have direct and inverse trigonometric functions, there is a variable x-photon and several trivial operations. The first thing that comes to mind (at least to me) is to look at the graphs of the functions ArcSin [Cos [x]] and ArcCos [Sin [x]]:





The given graphs already remind us very much of the “quadratic cosine” we need, but something is missing, it turns out that there is not enough “entanglement”, what we did is in fact the first level entanglement, but this is not enough, you need these two functions build, coming to the confusion of the second level. After several experiments with the available trivial operations, I stopped at the division and this is what happened (Fig. 4):

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It was here that I realized that I was not lost in the entanglement of the x-photon and everything just clears up.

It would seem that half the problem is solved and it remains stupid to copy the solution into two formulas of the form:







But I wanted to present everything with one formula and the search continued ...



Therefore it was necessary to analyze the graph presented in Fig.4. What is remarkable about it?

First, half the square is present, but you need to get rid of these ascending lines. How to achieve this? Only “annihilation”, that is, self-destruction of opposites. And it is here that we will need a module so that we have smooth, ascending and descending symmetrical lines. Therefore, I considered the following schedule:







It would seem that a small difference is a module, but a big difference - now we have symmetric (relative to the origin) ascending and descending lines, which are sufficiently “folded” and they will turn into a square ... But you don’t need to add them;







Q.E.D.



This feature

y=ArcSin[Cos[x]]/ArcCos[Abs[Sin[x]]] 

I called the “ white function ” because it is as perfect and harmonious as the white color. The white function is a complex function model of a quantum-entangled pair of argument x with itself. The white function also defines a whole class of like trigonometric functions of the form

 y=ArcSin[f1[x]]/ArcCos[Abs[f2[x]]] 

for example, this view can also include the function

 y=ArcSin[1/Tan[x]]/ArcCos[Abs[Tan[x]]] 

etc.



Sources in Wolfram Mathematica format - yadi.sk/d/3pl0lZMH3PzxCU

To estimate the accuracy, I decided to calculate for the white function the area under the graph in the vicinity of the point Pi / 2 from the negative side. And that's what happened:

 Evaluate[Integrate[ ArcSin[Cos[x]]/ArcCos[Abs[Sin[x]]], {x, N[Pi, 100]/2 - 0.000000000000000000001`100, N[Pi, 100]/2}]] 

Here I think they will consider me a complete ignoramus, but I will still say what it means.

Obviously, by the result in the limit, the real part tends to infinity, and the imaginary to zero, that is, ∞ + 0.0 * i

What can this mean? What is the point in the real part and imaginary? I think the real part is proportional to the accuracy

(which is easily seen by increasing the number of zeros in the formula above to get closer to the point Pi / 2), and the imaginary part is proportional to the error of calculation and this error tends to zero in the limit.



Therefore, we conclude that the found formula of the white function makes it possible to calculate with controlled accuracy, which is good news. But how is this possible? I have no answer - think with your head.



The problem in the comments is considered in sufficient detail, but the Pi / 2 point occupies a special place in the question of the white function, the tungsten painting draws it like this:



But you need to understand that this is an approximation, ideally in the limit there is a vertical line and there is no gap:



reddened the fact that the computer can not calculate!

Interestingly, at the junction points of blue and red, y = + - Sqrt [2] / 2

Abs [Cos [x]] / Cos [x]



using the Fourier series



Obviously, these methods are not comparable with the solution found in terms of accuracy and performance; there is no need to add hundreds of cosines, as the Fourier series does.



PS The comments found another solution with the help of complex numbers and the number E: Cos [Arctan [E ^ (I * x)]] / Sin [Arcctg [E ^ (I * x)]] , which with great accuracy gives the solution.

Modeling forms using the function found

A trivial example, a cube:

 a[x_] := ArcSin[Cos[x]]/ArcCos[Abs[Sin[x]]]; (* *) b[y_] := ArcSin[Cos[y]]/ArcCos[Abs[Sin[y]]]; (* *) c[z_] := ArcSin[Cos[z]]/ArcCos[Abs[Sin[z]]]; (* *) f[x_, y_] := a[x]*b[y]; time[t_] := c[t]; z = Table[ Plot3D[10*move*(1 + f[x, y]*time[move]), {x, -3*Pi/2, 3*Pi/2}, {y, -3*Pi/2, 3*Pi/2}, PlotRange -> {-50, 50}], {move, -Pi/2 + 0.1, Pi/2 - 0.1, Pi/100}]; z = Join[z, Reverse[z]]; Export["C:\\out.gif", z, "AnimationRepetitions" -> Infinity]