# Homework Help: Fourier transform of a compicated function

1. Sep 30, 2010

### sphys

Hi

Could someone help me to calculate the fourier transform of the following function:

rect(x/d)exp(2ipia|x|)

2. Sep 30, 2010

### mathman

What is rect(x/d)?

3. Oct 5, 2010

### sphys

rect(x/d) is a rectangle function.
rect(x/d)=1 if -d/2<x<d/2;
rect(x/d)=0 if x<-d/2 or x>d/2.

4. Oct 5, 2010

### mathman

It looks like a straightforward integration.
∫(x=-d/2,0)exp(-2πiax+itx) dx + ∫(x=0,d/2)exp(2πiax+itx) dx

5. Oct 5, 2010

### sphys

Instead of numerical solution, is there an analytical solution for this problem?

6. Oct 5, 2010

### vela

Staff Emeritus
That's not a numerical solution.

You could try using the convolution theorem to find the Fourier transform, but that seems like even more work.

7. Oct 5, 2010

### sphys

Can you please give me the solution using the convolution theorem?

8. Oct 6, 2010

### vela

Staff Emeritus
No, that's against the forum rules. Conceptually, it's straightforward. The convolution theorem tells you

$$\mathcal{F}[f(x)g(x)]=\mathcal{F}[f(x)]*\mathcal{F}[g(x)]$$

so you just have to find the transforms of the rectangle and exponential functions individually and convolve the results.