1. The problem statement, all variables and given/known data
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
2. Relevant equations
So far I've done a) and b) and the delta function turns out to be [itex] \delta (x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{i\omega x}d\omega [/itex]
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.
3. 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.
[tex]
\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
[/tex]
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.
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
2. Relevant equations
So far I've done a) and b) and the delta function turns out to be [itex] \delta (x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{i\omega x}d\omega [/itex]
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.
3. 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.
[tex]
\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
[/tex]
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.
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