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I have seen the usual method of using polar coordinates and double integration but is there a much simpler way to derive the value of the gaussian integral?

Krish Barwar
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  • Hi, welcome to Math SE. There are many ways to evaluate that integral, but the others might not be simpler in your opinion. This proof doesn't need a double integral. – J.G. Feb 17 '25 at 16:59
  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Feb 17 '25 at 17:00

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Possibly the most concise way.$$\mathcal I=2\int_0^\infty e^{-x^2}dx\stackrel{u=x^2}{=}\int_0^\infty u^{-1/2}e^{-u}du=\Gamma(\frac12)$$

Antony Theo.
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