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!

Some complex Fourier-like infinite integral

  1. Mar 11, 2007 #1
    1. The problem statement, all variables and given/known data
    The following integral is given:
    [tex]\int_{0}^{\infty} e^{ikx} dx[/tex]
    k & x are real.

    2. Relevant equations
    We know that:
    [tex]\int_{-\infty}^{\infty} e^{ikx} dx=2\pi \delta(k)[/tex]

    3. The attempt at a solution
    The answer is:
    [tex]\mathcal{I}=\pi \delta(k) + i \frac{1}{k}[/tex]
    I could easily prove the real part with the formula in 2, but couldn't be persuaded why the imaginary part is so (it barely makes sense). I tried:
    [tex]\Im \int_{0}^{\infty} e^{ikx} dx = \int_{0}^{\infty} =\sin(kx) dx = \lim_{M\rightarrow \infty} \frac{1}{k}(-\cos(kM)+1)[/tex]
    But [itex]\lim_{M\rightarrow \infty} \cos(kM)[/itex] is quite meaningless and I can't see why the whole expression should be k^{-1}...

    I'll appreciate any help,
    Last edited: Mar 11, 2007
  2. jcsd
  3. Mar 11, 2007 #2
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook