The discussion revolves around the identity involving the Gaussian integral: ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2##. Participants explore its usefulness, notation issues, and potential generalizations. The scope includes mathematical reasoning and technical clarification.
Participants express differing opinions on the usefulness of the identity, with some suggesting it may not be significant while others propose it could be generalized. There is no consensus on the identity's value or clarity due to notation issues.
There are unresolved issues regarding the notation and potential sign errors in the identity. The discussion reflects uncertainty about the implications of the identity and its applications.
IMO, no. Unless I'm missing something, this looks similar to the integral ##\int_{-\infty}^\infty e^{-x^2}dx## is evaluated. IOW, by instead looking at the double integral. I'd bet this technique is in most calculus textbooks, although usually as the integral ##\int_{-\infty}^\infty \int_{-\infty}^\infty e^{(-x^2 - y^2)/2}dy dx##.MevsEinstein said:Summary:: What the title says
I found this identity: ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2## by solving the integral of ##x*e^{-x^2}## and then finding its integration-by-parts equivalent. Is this identity useful at all?
There aren't any infinities in the formula.Mark44 said:IMO, no. Unless I'm missing something, this looks similar to the integral ##\int_{-\infty}^\infty e^{-x^2}dx## is evaluated. IOW, by instead looking at the double integral. I'd bet this technique is in most calculus textbooks, although usually as the integral ##\int_{-\infty}^\infty \int_{-\infty}^\infty e^{(-x^2 - y^2)/2}dy dx##.
I don't know how to fix that problem. I'm still 13.mathman said:Fix notation. Using 'x' too much. For example dxdx?
Use different letters for different things.MevsEinstein said:I don't know how to fix that problem. I'm still 13.
Should readMevsEinstein said:I found this identity: ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2## by solving the integral of ##x*e^{-x^2}## and then finding its integration-by-parts equivalent. Is this identity useful at all?
I couldn't edit the OP.drmalawi said:looks like also a sign error.