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.
The Gaussian Integral, also known as the error function, is a mathematical function that describes the area under the bell curve of a normal distribution. It is commonly used in statistics and probability calculations.
The Gaussian Integral has many applications in scientific research, particularly in fields such as physics, engineering, and economics. It is used to calculate probabilities, solve differential equations, and model natural phenomena.
Yes, the Gaussian Integral is a valuable tool for solving real-world problems that involve normal distributions. It is often used in data analysis, risk assessment, and optimization problems.
While the Gaussian Integral is a powerful mathematical tool, it does have some limitations. It can only be used for normal distributions and may not accurately represent non-normal data. Additionally, it may be computationally intensive for complex problems.
There are many resources available for learning about the Gaussian Integral, including textbooks, online tutorials, and scientific articles. Additionally, consulting with a mathematician or statistician can provide further insight into its uses and limitations.