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I want to change variables in the following integral:

Integrate[q*D[z[x, y],x]^2, {x, -∞, ∞}, {y, -∞, ∞}]

as follows:

{x == k1*x1, y == k1*y1}

and z[x,y]->k2*u[x1,y1]

That is what I tried:

Simplify[
  IntegrateChangeVariables[
   Inactive[Integrate][
    q*D[z[x, y], 
      x]^2, {x, -∞, ∞}, {y, -∞,
∞}], {x1, y1}, {x == k1*x1, y == k1*y1}], {x1 ∈ 
    Reals, y1 ∈ Reals, k1 > 0}] /. z -> (k2*u[#1/k1, #2/k1] &)

with the following effect:

enter image description here

This result is correct. However, my questions are

  1. Where the {True, True} comes from and what does it mean in this context?
  2. Is it eventually possible to remove it?
  3. Is it possible to change variables z->u within the function IntegrateChangeVariables, rather than as I did it?
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  • 1
    $\begingroup$ The {True, True} comes from simplifying the 3rd argument in the code returned by IntegrateChangeVariables[ Inactive[Integrate][ q*D[z[x, y], x]^2, {x, -\[Infinity], \[Infinity]}, {y, -\[Infinity], \[Infinity]} ], {x1, y1}, {x == k1*x1, y == k1*y1}]. That code seems erroneous to me (a bug, I mean). But the problem is better shown without the Simplify, imo. $\endgroup$
    – Goofy
    Commented Dec 24, 2023 at 3:05

1 Answer 1

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The following works well in 13.3.1 on Windows 10.

Assuming[k1 > 0, IntegrateChangeVariables[ Inactive[Integrate]
[ q*D[z[x, y],  x]^2, {x, -\[Infinity], \[Infinity]}, {y, -\[Infinity], 
\[Infinity]}], {x1, y1}, {x == k1*x1, y == k1*y1}]]

k1^2*q*Integrate[Derivative[1, 0][z][ k1*x1, k1*y1]^2, {x1, -Infinity, Infinity}, {y1, -Infinity, Infinity}]

Compare with a case of a specified z[x,y]

Assuming[k1 > 0, IntegrateChangeVariables[Inactive[Integrate][q*D[Exp[-x^4 - y^2]
  , x]^2, {x, -\[Infinity], \[Infinity]}, {y, -\[Infinity], \[Infinity]}], {x1, y1}, 
{x == k1*x1, y == k1*y1}]]

Inactive[Integrate][(16*k1^8*q*x1^6)/ E^(2*k1^2*(k1^2*x1^4 + y1^2)), {x1, -Infinity, Infinity}, {y1, -Infinity, Infinity}]

One sees problems with the derivative in the general case.

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2
  • $\begingroup$ Thank you. Do you know the answer to my first question? I would like to understand where this {True,True} comes from? $\endgroup$ Commented Nov 24, 2023 at 11:03
  • $\begingroup$ @AlexeiBoulbitch: No, I don't know answer to your first question. Ask Mathematica developers that. $\endgroup$
    – user64494
    Commented Nov 24, 2023 at 12:58

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