Is this identity containing the Gaussian Integral of any use?

In summary, this identity may not be very useful because it looks similar to an equation that is already known.
  • #1
MevsEinstein
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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?
 
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  • #2
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?
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##.
 
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  • #3
Fix notation. Using 'x' too much. For example dxdx?
 
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  • #4
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##.
There aren't any infinities in the formula.
 
  • #5
mathman said:
Fix notation. Using 'x' too much. For example dxdx?
I don't know how to fix that problem. I'm still 13.
 
  • #6
MevsEinstein said:
I don't know how to fix that problem. I'm still 13.
Use different letters for different things.
 
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  • #7
MevsEinstein 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?
Should read
##x\int e^{-x^2} dx - \int \int e^{-y^2} dydx = -e^{-x^2}/2## looks like also a sign error.
These "identities" can be generalized to many functions, its just partial integration and change of variables, for instance consider xsin(x2)
 
  • #8
$$\int xe^{-x^2}dx=-e^{-x^2}/2$$.
 
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  • #9
drmalawi said:
looks like also a sign error.
I couldn't edit the OP.
 

Related to Is this identity containing the Gaussian Integral of any use?

1. What is the Gaussian Integral?

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.

2. How is the Gaussian Integral used in scientific research?

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.

3. Can the Gaussian Integral be used to solve real-world problems?

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.

4. Are there any limitations to using the Gaussian Integral?

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.

5. How can I learn more about the Gaussian Integral and its applications?

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.

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