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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
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