How Do You Compute the Fourier Transform of x*f(x)?

[paxprobellum]

Homework Statement

I'm trying to do the Fourier Transform of "x*f(x)". According to some web sites (http://everything2.com/index.pl?node=fourier transform), it is simply a "property of integration", but I've having some issues.

Homework Equations

I'm using the form:
1/2pi int ( x f(x) e ^ (iwx) dx )

The Attempt at a Solution

I've tried doing it by parts, but I can't get it to work. My two attempts were:
1) u=f(x) and dV = x e^(iwx) dx
and
2) u=x*f(x) and dV = e^(iwx) dx

Using 1) I got -1/w^2 dF/dW
Using 2) I got (1/iw)[-int (e^(iwx) x f'(x) dx) - F(w)]

I know the answer is -i dF/dw (as it is in the Fourier transform table), but I can't seem to get there. =/

 

[mighty2000]

Thank you for your post. The Fourier transform of x*f(x) can indeed be derived using integration by parts, as you have attempted. However, there are some issues with your attempts that may have led to incorrect results. Here is a step-by-step guide on how to properly perform the Fourier transform of x*f(x):

1. Start with the integral 1/2pi int ( x f(x) e ^ (iwx) dx ).

2. Use integration by parts, with u=x and dV=f(x)e^(iwx)dx. This gives us:

1/2pi [x * F(x) e^(iwx)] - 1/2pi int (F(x) e^(iwx) dx), where F(x) is the Fourier transform of f(x).

3. Now, we need to solve the remaining integral, 1/2pi int (F(x) e^(iwx) dx). To do this, we can use the Fourier transform property e^(iwx) F(w). This means that the integral becomes:

1/2pi e^(iwx) F(w) dx

4. Plugging this back into our original integral, we get:

1/2pi [x * F(x) e^(iwx)] - 1/2pi e^(iwx) F(w) dx

5. Simplifying this further, we get:

1/2pi e^(iwx) [x * F(x) - F(w)]

6. Finally, taking the inverse Fourier transform of this expression, we get:

1/2pi int (e^(-iwx) [x * F(x) - F(w)] dx)

7. This simplifies to -i dF/dw, as expected.

I hope this helps you with your problem. Please let me know if you have any further questions or concerns. Keep up the good work!