Convergence problem for Fourier Transform of 1/t

[chingkui]

Hi,

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?

 

[mighty2000]

Hi there,

Thank you for your question. The Fourier Transform of f(t)=1/t is indeed known as the Hilbert Transformer. To show convergence of the integral of cos(t)/t around t=0, we can use the Dirichlet test for integrals. This test states that if f(t) is a continuous function on [a,b] and g(t) is a monotonic function on [a,b] with g(t) approaching 0 as t approaches infinity, then the integral of f(t)g(t) converges. In this case, we can let f(t)=cos(t) and g(t)=1/t. Since g(t) approaches 0 as t approaches infinity, the integral will converge if cos(t) is continuous on [0, infinity). This is indeed the case, as cos(t) is continuous on all real numbers. Therefore, the integral of cos(t)/t around t=0 will converge and the Fourier Transform of f(t)=1/t will be as you have stated: 1 for w>0, -1 for w<0, and 0 for w=0.

I hope this helps and provides a decent proof for the convergence of the integral. Keep up the good work in your studies!