Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By johnsomeone

Math Help - Computing an integral

  1. #1
    Member
    Joined
    Aug 2009
    Posts
    170
    Thanks
    8

    Computing an integral

    Hello, another question from an exam. It's a straight-forward computation, but I get stuck at the actual integration.

    Compute \int_{-\infty}^{\infty} e^{-x^2} \text{cos} 2kx \text{d}x, k>0 by integrating e^{-z^2} around the rectangle with vertices \pm a, \pm a + bi

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    147

    Re: Computing an integral

    z = x + iy \Rightarrow e^{-z^2} = e^{y^2-x^2} e^{-2xyi}

    \oint  e^{-z^2} dz = Countercockwise integral along rectangle with verticies \pm a, \pm a + bi

    = \int_{-a}^{a} e^{-x^2} dx + \int_{0}^{b} e^{y^2-a^2} e^{-2ayi} dy - \int_{-a}^{a} e^{b^2-x^2} e^{-2bxi}dx - \int_{0}^{b} e^{y^2-a^2} e^{2ayi} dy

    ...

    = \left( \int_{-a}^{a} e^{-x^2} dx - e^{b^2} \int_{-a}^{a} e^{-x^2} \cos(2bx) dx \right) + \ i \left( -2e^{-a^2} \int_{0}^{b} e^{y^2} \sin(2ay) dy + e^{b^2}\int_{-a}^{a} e^{-x^2} \sin(2bx) dx \right)

    That's what I got - I certainly could've messed up somewhere.

    Assuming that's correct, from that, it's possible to quickly find the answer.
    Last edited by johnsomeone; September 21st 2012 at 05:15 AM.
    Thanks from Bingk
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. fourier transform and integral computing
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: December 11th 2011, 01:07 AM
  2. Computing Integral
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 2nd 2011, 09:03 AM
  3. Computing an Integral
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 29th 2010, 07:49 AM
  4. Computing the Definite integral
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 21st 2009, 03:30 AM
  5. [SOLVED] computing integral
    Posted in the Calculus Forum
    Replies: 6
    Last Post: November 20th 2007, 08:45 AM

Search Tags


/mathhelpforum @mathhelpforum