# What does "{True,True}" mean in the result of IntegrateChangeVariables?

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:

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?
• 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. Commented Dec 24, 2023 at 3:05

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.

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