0
$\begingroup$

Starting from a function $\theta0[x\_ , y\_ , z\_ ]$, the terms $\varphi[x\_ , y\_ , z\_ ]$ constitutes a sequence of functions that is constructed as follows:

Order 1 (one function and its reflection) $$\varphi1[x\_ , y\_ , z\_ ]=\mathcal{H[z-z1]}(\alpha \theta0[x,y,z]) + \mathcal{H[-(z-z1)]}(\alpha \theta0[x,y,2z1-z])$$ $$\varphi1 R[x\_ , y\_ , z\_ ]= \varphi1[x,y,-z]$$

\[CurlyPhi]1[x_, y_, z_] := 
 HeavisideTheta[z - z1]*(\[Alpha]*\[Theta]0[x, y, z]) + 
  HeavisideTheta[-(z - z1)]*(\[Alpha]*\[Theta]0[x, y, 2*z1 - z])
\[CurlyPhi]1R[x_, y_, z_] := \[CurlyPhi]1[x, y, -z]

Higher orders use the results of previous one

Order 2 - use the results of order 1 $\varphi 2$ is obtained by substituting in $\varphi 1$, $\theta0 \rightarrow \varphi 1R$ so that is $$\varphi 2[x\_ , y\_ , z\_ ]=\mathcal{H[z-z1]}(\alpha \varphi 1R[x,y,z]) + \mathcal{H[-(z-z1)]}(\alpha \varphi 1R[x,y,2z1-z])$$ $$\varphi 2R[x\_ , y\_ , z\_ ]= \varphi 2[x,y,-z]$$

\[CurlyPhi]2[x_, y_, z_] := 
 HeavisideTheta[z - z1]*(\[Alpha]*\[CurlyPhi]1R[x, y, z]) + 
  HeavisideTheta[-(z - z1)]*(\[Alpha]*\[CurlyPhi]1R[x, y, 2*z1 - z])
\[CurlyPhi]2R[x_, y_, z_] := \[CurlyPhi]2[x, y, -z]

and so on...

\[CurlyPhi]3[x_, y_, z_] := 
 HeavisideTheta[z - z1]*(\[Alpha]*\[CurlyPhi]2R[x, y, z]) + 
  HeavisideTheta[-(z - z1)]*(\[Alpha]*\[CurlyPhi]2R[x, y, 2*z1 - z])
\[CurlyPhi]3R[x_, y_, z_] := \[CurlyPhi]3[x, y, -z]
\[CurlyPhi]4[x_, y_, z_] := 
 HeavisideTheta[z - z1]*(\[Alpha]*\[CurlyPhi]3R[x, y, z]) + 
  HeavisideTheta[-(z - z1)]*(\[Alpha]*\[CurlyPhi]3R[x, y, 2*z1 - z])

The final goal is to obtain the sum of $\theta0[x\_ , y\_ , z\_ ]$ and the terms $\varphi[x\_ , y\_ , z\_ ]$ in which only the function $\theta0[x\_ , y\_ , z\_ ]$ appears. An example output truncated at order 4 is given by: 

\[Theta]0[x, y, z] + \[CurlyPhi]1[x, y, z] + \[CurlyPhi]1R[x, y, 
  z] + \[CurlyPhi]2[x, y, z] + \[CurlyPhi]2R[x, y, z] + \[CurlyPhi]3[
  x, y, z] + \[CurlyPhi]3R[x, y, z] + \[CurlyPhi]4[x, y, z]

producing

\[Alpha] HeavisideTheta[-z - z1] \[Theta]0[x, y, -z] + \[Theta]0[x, y,
   z] + \[Alpha] HeavisideTheta[z - z1] \[Theta]0[x, y, 
   z] + \[Alpha] HeavisideTheta[-z + z1] \[Theta]0[x, 
   y, -z + 2 z1] + \[Alpha] HeavisideTheta[-z - 
    z1] (\[Alpha] HeavisideTheta[z - z1] \[Theta]0[x, y, 
      z] + \[Alpha] HeavisideTheta[-z + z1] \[Theta]0[x, 
      y, -z + 2 z1]) + \[Alpha] HeavisideTheta[z + z1] \[Theta]0[x, y,
    z + 2 z1] + \[Alpha] HeavisideTheta[
   z - z1] (\[Alpha] HeavisideTheta[-z - z1] \[Theta]0[x, 
      y, -z] + \[Alpha] HeavisideTheta[z + z1] \[Theta]0[x, y, 
      z + 2 z1]) + \[Alpha] HeavisideTheta[-z + 
    z1] (\[Alpha] HeavisideTheta[z - 3 z1] \[Theta]0[x, y, 
      z - 2 z1] + \[Alpha] HeavisideTheta[-z + 3 z1] \[Theta]0[x, 
      y, -z + 4 z1]) + \[Alpha] HeavisideTheta[-z - 
    z1] (\[Alpha] HeavisideTheta[
      z - z1] (\[Alpha] HeavisideTheta[-z - z1] \[Theta]0[x, 
         y, -z] + \[Alpha] HeavisideTheta[z + z1] \[Theta]0[x, y, 
         z + 2 z1]) + \[Alpha] HeavisideTheta[-z + 
       z1] (\[Alpha] HeavisideTheta[z - 3 z1] \[Theta]0[x, y, 
         z - 2 z1] + \[Alpha] HeavisideTheta[-z + 3 z1] \[Theta]0[x, 
         y, -z + 4 z1])) + \[Alpha] HeavisideTheta[
   z + z1] (\[Alpha] HeavisideTheta[-z - 3 z1] \[Theta]0[x, 
      y, -z - 2 z1] + \[Alpha] HeavisideTheta[z + 3 z1] \[Theta]0[x, 
      y, z + 4 z1]) + \[Alpha] HeavisideTheta[
   z - z1] (\[Alpha] HeavisideTheta[-z - 
       z1] (\[Alpha] HeavisideTheta[z - z1] \[Theta]0[x, y, 
         z] + \[Alpha] HeavisideTheta[-z + z1] \[Theta]0[x, 
         y, -z + 2 z1]) + \[Alpha] HeavisideTheta[
      z + z1] (\[Alpha] HeavisideTheta[-z - 3 z1] \[Theta]0[x, 
         y, -z - 2 z1] + \[Alpha] HeavisideTheta[z + 3 z1] \[Theta]0[
         x, y, z + 4 z1])) + \[Alpha] HeavisideTheta[-z + 
    z1] (\[Alpha] HeavisideTheta[
      z - 3 z1] (\[Alpha] HeavisideTheta[z - z1] \[Theta]0[x, y, 
         z] + \[Alpha] HeavisideTheta[-z + z1] \[Theta]0[x, 
         y, -z + 2 z1]) + \[Alpha] HeavisideTheta[-z + 
       3 z1] (\[Alpha] HeavisideTheta[z - 5 z1] \[Theta]0[x, y, 
         z - 4 z1] + \[Alpha] HeavisideTheta[-z + 5 z1] \[Theta]0[x, 
         y, -z + 6 z1])) + \[Alpha] HeavisideTheta[
   z + z1] (\[Alpha] HeavisideTheta[-z - 
       3 z1] (\[Alpha] HeavisideTheta[-z - z1] \[Theta]0[x, 
         y, -z] + \[Alpha] HeavisideTheta[z + z1] \[Theta]0[x, y, 
         z + 2 z1]) + \[Alpha] HeavisideTheta[
      z + 3 z1] (\[Alpha] HeavisideTheta[-z - 5 z1] \[Theta]0[x, 
         y, -z - 4 z1] + \[Alpha] HeavisideTheta[z + 5 z1] \[Theta]0[
         x, y, z + 6 z1])) + \[Alpha] HeavisideTheta[
   z - z1] (\[Alpha] HeavisideTheta[-z - 
       z1] (\[Alpha] HeavisideTheta[
         z - z1] (\[Alpha] HeavisideTheta[-z - z1] \[Theta]0[x, 
            y, -z] + \[Alpha] HeavisideTheta[z + z1] \[Theta]0[x, y, 
            z + 2 z1]) + \[Alpha] HeavisideTheta[-z + 
          z1] (\[Alpha] HeavisideTheta[z - 3 z1] \[Theta]0[x, y, 
            z - 2 z1] + \[Alpha] HeavisideTheta[-z + 3 z1] \[Theta]0[
            x, y, -z + 4 z1])) + \[Alpha] HeavisideTheta[
      z + z1] (\[Alpha] HeavisideTheta[-z - 
          3 z1] (\[Alpha] HeavisideTheta[-z - z1] \[Theta]0[x, 
            y, -z] + \[Alpha] HeavisideTheta[z + z1] \[Theta]0[x, y, 
            z + 2 z1]) + \[Alpha] HeavisideTheta[
         z + 3 z1] (\[Alpha] HeavisideTheta[-z - 5 z1] \[Theta]0[x, 
            y, -z - 4 z1] + \[Alpha] HeavisideTheta[
            z + 5 z1] \[Theta]0[x, y, 
            z + 6 z1]))) + \[Alpha] HeavisideTheta[-z + 
    z1] (\[Alpha] HeavisideTheta[
      z - 3 z1] (\[Alpha] HeavisideTheta[
         z - z1] (\[Alpha] HeavisideTheta[-z - z1] \[Theta]0[x, 
            y, -z] + \[Alpha] HeavisideTheta[z + z1] \[Theta]0[x, y, 
            z + 2 z1]) + \[Alpha] HeavisideTheta[-z + 
          z1] (\[Alpha] HeavisideTheta[z - 3 z1] \[Theta]0[x, y, 
            z - 2 z1] + \[Alpha] HeavisideTheta[-z + 3 z1] \[Theta]0[
            x, y, -z + 4 z1])) + \[Alpha] HeavisideTheta[-z + 
       3 z1] (\[Alpha] HeavisideTheta[
         z - 5 z1] (\[Alpha] HeavisideTheta[z - 3 z1] \[Theta]0[x, y, 
            z - 2 z1] + \[Alpha] HeavisideTheta[-z + 3 z1] \[Theta]0[
            x, y, -z + 4 z1]) + \[Alpha] HeavisideTheta[-z + 
          5 z1] (\[Alpha] HeavisideTheta[z - 7 z1] \[Theta]0[x, y, 
            z - 6 z1] + \[Alpha] HeavisideTheta[-z + 7 z1] \[Theta]0[
            x, y, -z + 8 z1])))

My main issue is how to properly write this sequence of function.

Very naively, I first tried to simplify the expression by using the substitution operator, so that for example the second term can be written by using the replacement

\[CurlyPhi]2[x_, y_, z_] := 
 HeavisideTheta[z - z1]*(\[Alpha]*\[Theta]0[x, y, z]) + 
   HeavisideTheta[-(z - z1)]*(\[Alpha]*\[Theta]0[x, y, 
       2*z1 - z]) /. {\[Theta]0 -> \[CurlyPhi]1R}

This might reduce the number of typos, but it is basically the same thing as copying and pasting and it does not get me very far.

Then I understood that I have to add an additional argument to the function to describe the order. As a newbie to programming, what I did was to define the symbol to be substituted as an argument.

\[CurlyPhi]0R = \[Theta]0;
\[CurlyPhi]symbol[\[CurlyPhi]order_, x_, y_, z_] := 
 HeavisideTheta[z - z1]*(\[Alpha]*\[CurlyPhi]order[x, y, z]) + 
  HeavisideTheta[-(z - z1)]*(\[Alpha]*\[CurlyPhi]order[x, y, 2*z1 - z])
\[CurlyPhi]Rsymbol[\[CurlyPhi]0, x_, y_, z_] := \[Theta]0[x, y, z]
\[CurlyPhi]Rsymbol[\[CurlyPhi]order_, x_, y_, z_] := \[CurlyPhi]order[
  x, y, -z]

That allows to construct the two terms of the sequence $\varphi[x\_ , y\_ , z\_ ]$ and $\varphi R[x\_ , y\_ , z\_ ]$ separately with \[CurlyPhi]symbol and \[CurlyPhi]Rsymbol, respectively, and I can use a table to write them:

Table[\[CurlyPhi]Rsymbol[
  Symbol["\[CurlyPhi]" <> ToString@(order - 1)], x, y, z], {order, 1, 
  4}]

Table[\[CurlyPhi]symbol[
  Symbol["\[CurlyPhi]" <> ToString@(order - 1) <> "R"], x, y, 
  z], {order, 1, 4}]

The produced output is indeed the symbolic required definition of each element separately.

Using \[CurlyPhi]symbol, for $\varphi[x\_ , y\_ , z\_ ]$ we have the following list, each element describing $\varphi 1$,$\varphi 2$,$\varphi 3$,$\varphi 4$

{\[Alpha] HeavisideTheta[z - z1] \[Theta]0[x, y, 
    z] + \[Alpha] HeavisideTheta[-z + z1] \[Theta]0[x, 
    y, -z + 2 z1], \[Alpha] HeavisideTheta[z - z1] \[CurlyPhi]1R[x, y,
     z] + \[Alpha] HeavisideTheta[-z + z1] \[CurlyPhi]1R[x, 
    y, -z + 2 z1], \[Alpha] HeavisideTheta[z - z1] \[CurlyPhi]2R[x, y,
     z] + \[Alpha] HeavisideTheta[-z + z1] \[CurlyPhi]2R[x, 
    y, -z + 2 z1], \[Alpha] HeavisideTheta[z - z1] \[CurlyPhi]3R[x, y,
     z] + \[Alpha] HeavisideTheta[-z + z1] \[CurlyPhi]3R[x, 
    y, -z + 2 z1]}

and using \[CurlyPhi]Rsymbol for $\varphi R[x\_ , y\_ , z\_ ]$ we have the following list, each element describing $\varphi R0$,$\varphi R1$,$\varphi R2$,$\varphi R3$

{\[Theta]0[x, y, z], \[CurlyPhi]1[x, y, -z], \[CurlyPhi]2[x, 
  y, -z], \[CurlyPhi]3[x, y, -z]}

However I got stuck at this point since I would not now how to map the proper association of the elements or perform the required substitutions so that I end up with a list of functions only $\theta0[x\_ , y\_ , z\_ ]$.

How would that be possible to perform the substitution?

Or, even better, what would be a more elegant version of constructing this sequence of functions? I feel that this is a textbook definition of some form of recursion but I cannot figure it out how to write it out.

Thank you all very much for sharing your knowledge and advice.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

I think that the following code should do what you want:

\[CurlyPhi][0, x_, y_, z_] := \[Theta]0[x, y, z];
\[CurlyPhi][n_, x_, y_, z_] := \[CurlyPhi][n, x, y, z] = Simplify[
   HeavisideTheta[z - z1] (\[Alpha] \[CurlyPhi]R[n - 1, x, y, z]) + 
   HeavisideTheta[-(z - z1)] (\[Alpha]*\[CurlyPhi]R[n - 1, x, y, 2 z1 - z])];
\[CurlyPhi]R[0, x_, y_, z_] := \[Theta]0[x, y, z];
\[CurlyPhi]R[n_, x_, y_, z_] := \[CurlyPhi][n, x, y, -z];
Improve this answer
$\endgroup$
3
  • $\begingroup$ Thank you very much - this is great! I would only add \[CurlyPhi]R[0, x_, y_, z_] := \[Theta]0[x, y, z]; as a definition otherwise for n=1 [CurlyPhi][1, x_, y_, z_] gives \[Alpha] (HeavisideTheta[z - z1] \[Theta]0[x, y, -z] + HeavisideTheta[-z + z1] \[Theta]0[x, y, z - 2 z1]) instead of what I should expect \[Alpha] (HeavisideTheta[z - z1] \[Theta]0[x, y, z] + HeavisideTheta[-z + z1] \[Theta]0[x, y, z - 2 z1]). Does it seem right to you? $\endgroup$
    – DanielAmmering
    May 30, 2019 at 12:26
  • 1
    $\begingroup$ @DanielAmmering Done! I was wondering why I didn't get expected results. $\endgroup$
    – Somos
    May 30, 2019 at 12:31
  • $\begingroup$ Thank you very much Somos - it is a great help to me :) $\endgroup$
    – DanielAmmering
    May 30, 2019 at 12:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.