What is the Fourier Transform of 1/t?

[DieCommie]

Homework Statement

Find the Fourier Transform of \frac {1}{t}

Homework Equations

Euler's equations I think...

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). I am pretty sure that is the way to go, but I can't seem to make it work. I think the answer is plus or minus i (from google searches), but I can't 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 don't 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, that's not what I want to hear!

We did some complex integration with poles in a different class. I didnt get it at all. I don't think that is required for this class. I am going to 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]

fourier sine transform of 1/sqrt x

can u please help me out with Fourier sine transform of 1/ sqrt x

 

[banhijit]

i need the solution asap...

 

[banhijit]

are u here?quasar987

 

[toneboy1]

ok, 1/t is like 1/w, if you times the numerator and denominator by j its like the Duality property (j* 1/(jt) ), so its like j2*pi*x(-w) = 2j*pi(-0.5+u(-w))
as you can see from the 1/jw transformation on the table.

 

[STEMucator]

This thread is like 4 years old. Why did you grave dig it?

 

[toneboy1]

Because I wanted to know the answer and I didn't think it had been adoquately addressed, because It hadn't, so I solved it for the next person to find it on google.