Integration test of dirac delta function as a Fourier integral

[Risborg]

Homework Statement

Problem:
a) Find the Fourier transform of the Dirac delta function: δ(x)
b) Transform back to real space, and plot the result, using a varying number of Fourier components (waves).
c) test by integration, that the delta function represented by a Fourier integral integrates to 1

Homework Equations

So far I've done a) and b) and the delta function turns out to be $\delta (x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{i\omega x}d\omega$

I've plotted this and it seems to be correct, and I also asked some other students in class and they got the same result, so i don't think that's the issue.

The Attempt at a Solution

So to solve c) I try to integrate δ(x) from -∞ to ∞, but it shouldn't really matter as long as 0 is between the integration limits.
$$\frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i\omega x}d\omega dx\\ \frac{1}{2\pi} \int_{-\infty}^{\infty}\left[\frac{1}{ix}e^{i\omega x}\right]_{\infty}^{\infty} dx\\ \frac{1}{\pi} \int_{-\infty}^{\infty}\frac{1}{x}\frac{1}{2i}(e^{i\infty x} - e^{-i\infty x}) dx\\ \frac{1}{\pi} \int_{-\infty}^{\infty}\frac{1}{x}\sin(\infty x) dx$$
Apparently I end up with an integral that's impossible to solve (without approximation), and the sine function has infinity as its argument... So I was hoping you would know where I went wrong.

 

[Ray Vickson]

Homework Statement

Problem:
a) Find the Fourier transform of the Dirac delta function: δ(x)
b) Transform back to real space, and plot the result, using a varying number of Fourier components (waves).
c) test by integration, that the delta function represented by a Fourier integral integrates to 1

Homework Equations

So far I've done a) and b) and the delta function turns out to be $\delta (x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{i\omega x}d\omega$

I've plotted this and it seems to be correct, and I also asked some other students in class and they got the same result, so i don't think that's the issue.

The Attempt at a Solution

So to solve c) I try to integrate δ(x) from -∞ to ∞, but it shouldn't really matter as long as 0 is between the integration limits.
$$\frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i\omega x}d\omega dx\\ \frac{1}{2\pi} \int_{-\infty}^{\infty}\left[\frac{1}{ix}e^{i\omega x}\right]_{\infty}^{\infty} dx\\ \frac{1}{\pi} \int_{-\infty}^{\infty}\frac{1}{x}\frac{1}{2i}(e^{i\infty x} - e^{-i\infty x}) dx\\ \frac{1}{\pi} \int_{-\infty}^{\infty}\frac{1}{x}\sin(\infty x) dx$$
Apparently I end up with an integral that's impossible to solve (without approximation), and the sine function has infinity as its argument... So I was hoping you would know where I went wrong.

The last three lines are nonsense; you need to take a limit. Use
$$\frac{1}{2\pi} \int_{-\infty}^{\infty}\int_N^N e^{i\omega x}d\omega dx$$
for finite $N > 0$. Work it through, then take the limit as $N \to \infty$.

Hint: Dirichlet Kernel.