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Feb 4, 2021 at 16:59 comment added Will Sawin @asma Essentially, yes. What's strange is the integral is defined in a purely formal way for generalized functions, so the usual change of variable formula doesn't exist, but I think the author has chosen to write this to look like the usual change of variable formula. It's somewhat irrelevant here because that term is always 1. I think for a noncompact group where the determinant was not always 1, you would need to put a determinant term in the first formula as well, to match this one.
Feb 4, 2021 at 8:21 vote accept Mira
Feb 4, 2021 at 8:20 comment added Mira thank you so much @Will Sawin for your answer ! I just still have a question , why the determinant appear in this formula, does it come from a change of variable ?
Feb 3, 2021 at 23:37 history answered Will Sawin CC BY-SA 4.0