1. Not finding help here? Sign up for a free 30min 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!

Complex Variable-definite integral

  1. Oct 14, 2008 #1
    Complex Variable---definite integral

    Show that
    f(k) = [tex]\frac{1}{2i\pi}[/tex] [tex]\int^{\infty}_{-\infty}[/tex] [tex]\frac{e^{ikx}}{x-i\epsilon}dx[/tex] =

    1, if k>0
    0, if k<0

    where [tex]\epsilon[/tex] > 0

    The attempt at a solution

    1st. step:
    Consider : [tex]\oint^{\infty}_{-\infty}[/tex] [tex]\frac{e^{ikz}}{z-i\epsilon}dz[/tex]

    the residule is : e^(-k[tex]\epsilon[/tex])

    so [tex]\int^{\infty}_{-\infty}[/tex] [tex]\frac{e^{ikx}}{x-i\epsilon}dx[/tex]

    = [tex]{2i\pi}[/tex] e^(-k[tex]\epsilon[/tex])

    [tex]\frac{1}{2i\pi}[/tex] [tex]\int^{\infty}_{-\infty}[/tex] [tex]\frac{e^{ikx}}{x-i\epsilon}dx[/tex]

    = e^(-k[tex]\epsilon[/tex])
  2. jcsd
  3. Oct 14, 2008 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    Re: Complex Variable---definite integral

    This makes no sense. What contour are you integrating over?

  4. Oct 14, 2008 #3
    Re: Complex Variable---definite integral

    um...the pole is at z= iε
    so i find the residule at z=iε
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Complex Variable-definite integral