Method NO. 3 : Neat Integral


Today, I decided to post a very eye-catching method for calculating an integral which shows very often in Statistical Mechanics in Physics.

Let’s try to calc this \int\limits_{-\infty}^{\infty}e^{-x^{2}} dx

First let’s mark it with a name K , then

K^{2} = \int\limits_{-\infty}^{\infty}e^{-x^{2}} dx \int\limits_{-\infty}^{\infty}e^{-y^{2}} dy = \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}e^{-(x^{2}+y^{2})} dx dy

What is just a double integral over the whole 2d surface. It can be also written in polar coordinate system as K^{2} = \int\limits_{0}^{\infty}e^{-r^{2}} \cdot 2\pi r dr  = \pi \cdot \int\limits_{0}^{\infty}e^{-t} dt = \pi

What means that K = \int\limits_{-\infty}^{\infty}e^{-x^{2}} dx = \sqrt{\pi}


7 Responses to “Method NO. 3 : Neat Integral”

  1. I think we studied this method in the faculty 😀
    Anyway, Well done 😉

  2. 2 the7new7ramanujan

    i was also fascinated by it when i saw it for the first time.

  3. maybe you need to give an explanation why the gaussian integral must be solved in polar cordinate, and also important to introduce the definition of gamma function.

  4. 4 masteranza

    I don’t think I’ll have to explain that – it’s just another way to represent a given function in which it’s easier to integrate.
    It’s just like using Cauchy instead Heine def of counting the limit.
    I don’t have any clue why I should introduce the def of gamma function either.

  5. Sorry, here I want to inform you all that in my website on this address, I post a new integral so-called Bernoulli Integral.
    Maybe useful for you. Thanks.

  6. can u brief how u integrate the 2nd surface, can’t u?

  7. Well, i’m integrating thin rings of radius r over all surface.

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