1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Science Advisor
    Homework Helper

    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:
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook