Struggling with Fourier Transform in Math Methods Class

[rdfloyd]

I haven't had differential equations yet, so I am struggling in your math methods class. I understand what a Fourier Transform is, but I'm having trouble with this particular problem.

Homework Statement

Here's a screenshot. Better than I can write it.

http://i.imgur.com/PQ6tB.png

The Attempt at a Solution

Here's what I did:

http://i.imgur.com/JuUzu.jpg

The capital letters have already been transformed, so if I take the inverse transformation, I should end up with what I had to begin with.

Where I get stuck is with the $Q, \frac{1}{D}, \frac{1}{(w^{2}+k^{2})}$. Is it possible to split the $\Delta[w]$ up from the fraction, because that would just be back to $\delta[x]$.

If I'm completely wrong and beyond hope, just tell me and I will go cry in a corner.

Thanks!

 

[SadScholar]

Oh hey, I think the issue may be that $\delta[x]$ is actually the dirac delta function, which has the property that
$$
\int_{-\infty}^\infty f(x)\delta(x)\, \mathrm{d}x = f(0)
$$
This would mean that
$$
\mathbb{F}[Q\delta(x)]=Q
$$
See if this fixes things. I tried the problem and still had a good deal of trouble with it , but you may be able to swing the rest from there.

PS: I'd be interested to see the rest of your solution when you get it. I suspect it may have to do with absolute values and or the step function.

 

[rdfloyd]

Using what you said (which makes sense; can't believe I didn't see that), I got this:

http://i.imgur.com/lBkuj.jpg

There were absolute values, however, I omitted them because I didn't think they were necessary.

 

[SadScholar]

Hmmm...
You know how $|x|$ has slope -1 until $x=0$, and then it has slope 1? I think you might be able to make this claim:
$$
\frac{\mathrm{d}|x|}{\mathrm{d}x}=2\mathbb{H}(x)-1\text{, where H is the Heaviside step function. Note also that}\\
\frac{\mathrm{d}\mathbb{H}}{\mathrm{d}x}=\delta(x)
$$
I have a suspicion this might be somehow related. I still haven't figured it out, but now it's starting to bother me.

 

[rdfloyd]

That's the first time I've heard of the Heaviside. What are it's uses (not only to this problem)?

 

[SadScholar]

I don't know! In fact, I don't even know why or where I heard of it! But check this page out, it's pretty cool: http://mathworld.wolfram.com/HeavisideStepFunction.html
I guess if you wanted to, you could also represent the T=0 Fermi distribution using the Heaviside... Not sure that would actually be useful, though.