Fourier Transform of 1/t

[DieCommie]

1. The problem statement, all variables and given/known data
Find the Fourier Transform of \frac {1}{t}



2. Relevant equations
Euler's equations I think...


3. The attempt at a solution
I tried splitting up the integral into two. One from -\inf to 0 and the other from 0 to \inf . Not much help there. I tried using e^{ix} = cos(x) + isin(x) . Im pretty sure that is the way to go, but I cant seem to make it work. I think the answer is plus or minus i (from google searches), but I cant make the steps to get there. Could someone give me some tips, or out line the steps? Thank you

[quasar987]

It would help to know that the integral from 0 to infinity of sinx/x is pi/2 !

[DieCommie]

Thank you, that does help. My teach. said dont use a table though... But this is better than nothing.

What is the integral from 0 to inf for cosx/x ?

[quasar987]

cos(x) = sin(x+pi/2)

[Dick]

cos(x) = sin(x+pi/2)

The integral of cos(x)/x from 0 to infinity just plain does not exist. As far as I know you can't do things like the fourier transform of 1/t by changing them into real integrals. You have to express them as contour integrals in the complex plane and pick a convergent contour or pull a residue theorem argument. Or do you know some trick I don't??

[quasar987]

No, I suppose you'Re right!

[DieCommie]

bah, thats not what I want to hear!

We did some complex integration with poles in a different class. I didnt get it at all. I dont think that is required for this class. Im gonna stick with the sinx/x = pi/2 unless somebody has a better idea.

[Dick]

If you've looked up the results then you should know that the integral of (1/t)*exp(i*t*x) depends on a discrete function of the value of x. That's a pretty sure sign that a contour choice is involved. Neglect this at your own risk.

[jmooney5115]

Hello. I am new to fourier transforms. Also I have not studied contour integration. In entry 309 in the table on wikipedia the answer to the fourier transform of 1/t = − i*pi*sgn(w).

The answer I get is i*pi*sgn(t). I'm not sure where the (-) comes from. I get, skipping a few steps: the integral with limits from -inf to inf of isin(wt)/t dt.

From my notes the integral from -inf to inf of sin(wt)/t would be = pi*sgn(w). I would assume when an imaginary number is in there you just treat it as a constant?

What am I missing here? Is my assumption wrong?

Thanks.

[asadzaman_007]

Hey,
Using Euler's formula, I'v found the FT of 1/(Pi*t) as -j. integration of cos(x)/x from -inf to inf is zero, as odd function. And using integration of sin(x)/x from -inf to inf = Pi. Using these two we easily can get FT of 1/(Pi.t) is equal to -j.
Using a known FT of rectangular(t/Tau) and X(0) or x(0) formulas of FT and IFT we can get the integration of sin(x)/x.

[banhijit]

can u plz help me out with fourier sine transform of 1/ sqrt x

[banhijit]

i need the solution asap...

[banhijit]

are u here?????quasar987