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Convergence problem for Fourier Transform of 1/t

  1. Feb 14, 2006 #1

    I am trying to derive the Fourier Transform of f(t)=1/t, but I have trouble showing the integral of exp(-jwt)/t converges near t=0 (for the real part). Essentially, I need to show convergence for the integral of cos(t)/t around t=0.
    I could show the complex part converges (i.e. integral of sin(wt)/t from -inf to inf) but don't know what it equals to. Have anyone seen that before?
    This function is called "Hilbert Transformer" and is used in some of my EE books (DSP, Communication, etc.), all these books claim that the Fourier Transform F{1/t}=1 for w>0, -1 for w<0, and 0 for w=0, but none of them gives any decent proof.
    Does this integral even converge?
  2. jcsd
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