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Integrate e^(-pi^2*x^2)

  1. Mar 8, 2008 #1
    How would I integrate e^(-pi^2*x^2) from -infinite to infinite? I know that the integral of e^(-x^2) from -infinite to infinite is sqr(pi), in which sqr is square root.
  2. jcsd
  3. Mar 8, 2008 #2


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    In general once can prove that
    [tex]\int_{-\infty}^{\infty} e^{-\alpha x^2} \, \mathrm{d}x = \sqrt{\frac{\pi}{\alpha}}, [/tex]
    for [itex]\alpha > 0[/itex] (and even [itex]\alpha \in \mathbb{C}, \operatorname{Re}(\alpha) > 0[/itex]).

    Then if you take [itex]\alpha = 1[/itex] or [itex]\alpha = \pi^2[/itex] you will get either of the integrals in your post.
  4. Mar 8, 2008 #3
    I see...thank you.
  5. Mar 8, 2008 #4
    Actually, I can see in this case that a=b so b-a=0 and therefore

    [tex]\int_a^b e^{-\pi x^2}\,dx \rightarrow \int_{-\infty}^{\infty} e^{-\pi x^2}\,dx=1[/tex]

    But that's probably beside the point. :smile:
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